archnes - DSpace@MIT

Unstable operations in the Bousfield-Kan spectral sequence

for simplicial commutative F2-algebras

by

Michael Jack Donovan

BCom with BSc, Macquarie University (2009)BSc (Hons), University of Sydney (2010)

Submitted to the Department of Mathematicsin partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

ARCHNESMASSACHUSETTS (NSTmTjTE

OF TECHNOLOLGY

JUN 3 02015

1 LIBRARIES

June 2015

@ Michael Jack Donovan, MMXV. All rights reserved.

The author hereby grants to MIT permission to reproduce and to distribute publiclypaper and electronic copies of this thesis document in whole or in part in any

medium now known or hereafter created.

Signature redactedA u th o r .. .. ... . ..................... ..... ....... ..... ...... .. ... .......... .........

Department of MathematicsApril 27, 2015

Certified by ...Signature redacted

Haynes MillerProfessor of Mathematics

Thesis Supervisor

Signature redactedA ccepted by .... ..........................

Alexei BorodinChairman, Department Committee on Graduate Theses

Unstable operations in the Bousfield-Kan spectral sequence for simplicial

commutative F2-algebras

by

Michael Jack Donovan

Submitted to the Department of Mathematicson April 27, 2015, in partial fulfillment of the

requirements for the degree ofDoctor of Philosophy

Abstract

In this thesis we study the Bousfield-Kan spectral sequence (BKSS) in the Quillen modelcategory sWom of simplicial commutative F2 -algebras. We develop a theory of unstable

operations for this BKSS and relate these operations with the known unstable operationson the homotopy of the target. We also prove a completeness theorem and a vanishing linetheorem which, together, show that the BKSS for a connected object of sWom convergesstrongly to the homotopy of that object.

We approach the computation of the BKSS by deriving a composite functor spectral

sequence (CFSS) which converges to the BKSS E2 -page. In fact, we generalize the con-

struction of this CFSS to yield an infinite sequence of CFSSs, with each converging to the

E2-page of the previous. We equip each of these CFSSs with a theory of unstable spectral

sequence operations, after establishing the necessary chain-level structure on the resolu-

tions defining the CFSSs. This technique may also yield operations on Blanc and Stover's

generalized Grothendieck spectral sequences in other settings.We are able to compute the Bousfield-Kan E2-page in the most fundamental case, that

of a connected sphere in sWom, using the structure defined on the CFSSs. We use this

computation to describe the Ei-page of a May-Koszul spectral sequence which converges

to the BKSS E2 -page for any connected object of sWom. We conclude by making two

conjectures which would, together, allow for a full computation of the BKSS for a connected

sphere in sWom.

Thesis Supervisor: Haynes MillerTitle: Professor of Mathematics

3

4

Acknowledgments

Firstly, I would like to thank my advisor Haynes Miller for his ongoing support and guid-

ance over the course of this Ph.D. Without doubt I learned more algebraic topology from my

meetings with Haynes than from all other sources combined, and at every meeting he con-

tributed materially to my intellectual development. This work owes much to his influence,

and to the ideas to be found in his Sullivan Conjecture paper [43].

Thanks to Paul Goerss, for visiting to take part in my examination committee, and

whose Asterisque volume [33j has had great influence on my work. Thanks too to Michael

Ching, Bill Dwyer, Benoit Fresse and John Harper, for helpful conversations on various

topics appearing in this thesis. I would also like to thank Michael Andrews for always being

available to answer questions and share ideas.

I would like to thank my wonderful classmates, in particular Michael Andrews, Nate

Bottman, Saul Glasman, Jiayong Li and Alex Moll, for sharing this experience with me.

Thanks too to Sonia and Eric and to Bella and Gilad - such wonderful friends as these are

truly rare.

Thanks to my mum and dad, and to Trish and Craig, for always encouraging me to

pursue my interests unreservedly, and for making the long trip from Australia to visit me

so often.

Finally, I would like to thank my wife Dana for her patience, love and support. Growing

together with Dana has been the most important and formative experience of the last five

years of my life.

5

6

Contents

1 Introduction

1.1 The classical Bousfield-Kan spectral sequence

1.2

1.3

1.4

1.5

1.6

1.7

1.8

11

. . . . . . . . . . . . . . . . . . 12

The various categories sC . . . . . . . . . . . . . . . . . .

The Bousfield-Kan spectral sequence in s'eom . . . . . . .

The first composite functor spectral sequence . . . . . . .

Higher composite functor spectral sequences . . . . . . . .

Computing with the composite functor spectral sequences

The Bousfield-Kan spectral sequence for S o" . . . . . . .

O verview . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . 1 5

. . . . . . . . . . . 15

. . . . . . . . . . . 19

. . . . . . . . . . . 2 1

. . . . . . . . . . . 23

. . . . . . . . . . . 24

. . . . . . . . . . . 24

2 Background and conventions

2.1 Universal algebras . . . . . . . . . . . . . . . .

2.2 The functor QC of indecomposables . . . . . . .

2.3 Quillen's model structure on sC and the bar cons

2.4 Categories of graded F2-vector spaces and linear

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

The Dold-Kan Correspondence . . . . . . . . .

Skeletal filtrations of almost free objects . . . .

Dold's Theorem . . . . . . . . . . . . . . . . . .

Homology and cohomology functors H2 and H*

The action of E2 on V2 . . . . . . . . . . . . .

Lie algebras in characteristic 2 . . . . . . . . .

Non-unital commutative algebras . . . . . . . .

First quadrant cohomotopy spectral sequences

Second quadrant homotopy spectral sequences

3 Homotopy operations and cohomology operatic

3.1 The spheres in sC and their mapping cones . .

3.2 Homotopy groups and C-il-algebras . . . . . . .

3.3 Cohomology groups and C-H*-algebras . . . . .

27

. . . . . . . . . . . . . . . . . 2 7

. . . . . . . . . . . . . . . . . 2 8

truction . . . . . . . . . . . . 28

dualization . . . . . . . . . . 29

. . . . . . . . . . . . . . . . . 3 0

. . . . . . . . . . . . . . . . . 3 3

. . . . . . . . . . . . . . . . . 3 4

. . . . . . . . . . . . . . . . . 3 4

. . . . . . . . . . . . . . . . . 3 5

. . . . . . . . . . . . . . . . . 3 7

. . . . . . . . . . . . . . . . . 3 9

. . . . . . . . . . . . . . . . . 4 0

. . . . . . . . . . . . . . . . . 4 1

ns 47

. . . . . . . . . . . . . . . . . 4 7

. . . . . . . . . . . . . . . . . 4 8

. . . . . . . . . . . . . . . . . 5 0

7

3.4

3.5

3.6

3.7

3.8

The reverse Adams spectral sequence . . . . . . . . . . .

The smash coproduct . . . . . . . . . . . . . . . . . . . .

Cofibrant replacement via the small object argument . .

Homology groups and C-H.-coalgebras . . . . . . . . . .

The Hurewicz map, primitives and homology completion

3.9 The smash product of homology coalgebras . . .

3.10 The quadratic part of a C-expression . . . . . . .

4 The Bousfield-Kan spectral sequence

4.1 Identification of El and E2 . . . . . . . . . . . .

4.2 The Adams tower . . . . . . . . . . . . . . . . . .

4.3 Connectivity estimates and homology completion

4.4 Iterated simplicial bar constructions . . . . . . .

5 Constructing homotopy operations

5.1 Higher simplicial Eilenberg-Mac Lane maps . . .

5.2 External unary homotopy operations . . . . . . .

5.3 External binary homotopy operations . . . . . . .

5.4

5.5

Homotopy operations for simplicial commutative a

Homotopy operations for simplicial Lie algebras .

. . . . . . . . . . . . 5 2

. . . . . . . . . . . . 5 2

. . . . . . . . . . . . 5 3

. . . . . . . . . . . . 5 6

. . . . . . . . . . . . 5 7

. . . . . . . . . . . . 5 9

. . . . . . . . . . . . 5 9

63

. . . . . . . . . . . . . . . . 63

. . . . . . . . . . . . . . . . 64

. . . . . . . . . . . . . . . . 67

. . . . . . . . . . . . . . . . 72

75

. . . . . . . . . . . . . . . . 75

. . . . . . . . . . . . . . . . 76

. . . . . . . . . . . . . . . . 77

lgebras . . . . . . . . . . . 78

. . . . . . . . . . . . . . . . 80

6 Constructing cohomology operations

6.1 Higher cosimplicial Alexander-Whitney maps . . . . . . . .

6.2 External unary cohomotopy operations . . . . . . . . . . . .

6.3 Linearly dual homotopy operations . . . . . . . . . . . . . .

6.4 External binary cohomotopy operations . . . . . . . . . . .

6.5 Chain level structure for cohomology operations . . . . . . .

6.6 Cohomology operations for simplicial commutative algebras

6.7 The categories W(0) and U(0) . . . . . . . . . . . . . . . . .

6.8 Cohomology operations for simplicial Lie algebras . . . . . .

85

85

86

86

87

88

90

92

94

7 Homotopy operations for partially restricted Lie algebras 99

7.1 The categories (n) of partially restricted Lie algebras . . . . . . . . . . . . . 99

7.2 Homotopy operations for s1(n) . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.3 The category U(n + 1) of unstable partial right A-modules . . . . . . . . . . . 101

7.4 The category W(nr+ 1) of Z(n)-H-algebras . . . . . . . . . . . . . . . . . . . . 101

7.5 The factorization QL(n) o QU(n) of QW(n) . . . . . . . . . . . . . . . . . . . . . 102

7.6 Decomposition maps for Z(n) and W(n) . . . . . . . . . . . . . . . . . . . . . 103

8

8 Operations on W(n)- and U(n)-cohomology

8.1 Vertical 6-operations on H, and H ) . . . . . . . . . . .

8.2 Vertical Steenrod operations for Hey and H when n > 1

8.3 Horizontal Steenrod operations and a product for H . . .

8.4 Relations between the horizontal and vertical operations . . .

8.5 The categories Mhv(n + 1) . . . . . . . . . . . . . . . . . . . .

8.6 Compressing sequences of Steenrod operations . . . . . . . . .

9 Koszul complexes calculating U(n)-homology

9.1 The Koszul complex and co-Koszul complex . . . . . . . . . .

9.2 The W(n + 1)-structure on H* X . . . . . . . . . . . . . . .

10 Operations on second quadrant homotopy spec

10.1

10.2

10.3

10.4

10.5

10.6

tral

Operations with indeterminacy . . . . . . . . .

Maps of mixed simplicial vector spaces . . . . .

An external spectral sequence pairing [text . . .

External spectral sequence operations Sqe .

External spectral sequence operations jext . . .

Internal operations on [EX] for X E csWom .

11 Operations in the Bousfield-Kan spectral sequence

11.1 An alternate definition of the Adams tower . . . . .

11.2 A modification of the functor R 1 . . . . . . . . . . .

11.3 Definition and properties of the BKSS operations . .

11.4 A chain-level construction *es inducing HC . . . . .

11.5 A three-cell complex with non-trivial bracket . . . .

11.6 A chain level construction of j*...... .....

11.7 A two-cell complex with non-trivial P' operation

11.8 A chain level construction of 0*.............

11.9 Proof of Proposition 11.2 . . . . . . . . . . . . . . . .

12 Composite functor spectral sequences

12.1 The factorization of QW(n) and resulting CFSSs . . .

12.2

12.3

12.4

12.5

12.6

The Blanc-Stover comonad in categories monadic ove

105

. . . . . . . . . 105

. . . . . . . . . 109

. . . . . . . . . 110

. . . . . . . . . 113

. . . . . . . . . 114

. . . . . . . . . 115

117

. . . . . . . . . 117

. . . . . . . . . 120

sequences 127

. . . . . . . . . . . . . . 128

. . . . . . . . . . . . . . 129

. . ... . . . . . . . . . ..129

. . . . . . . . . . . . . . 130

. . . . . . . . . . . . . . 133

. . . . . . . . . . . . . . 139

143

. . . . . . . . . . . . . . 143

. . . . . . . . . . . . . . 145

. . . . . . . . . . . . . . 147

. . . . . . . . . . . . . . 149

. . . . . . . . . . . . . . 15 1

. . . . . . . . . . . . . . 152

. . . . . . . . . . . . . . 154

. . . . . . . . . . . . . . 155

. . . . . . . . . . . . . . 156

161

. . . . . . . . . . . . . . 16 1

r F2 -vector spaces . . . . 162

A chain-level diagonal on the W9 construction . . . . . . . . . . . . . . .

Quadratic gradings in the CFSSs . . . . . . . . . . . . . . . . . . . . . .

The edge homomorphism and edge composite . . . . . . . . . . . . . . .

An equivalent reverse Adams spectral sequence . . . . . . . . . . . . . .

. . . 165

. . . 167

. . . 167

. . . 169

9

13 Operations in composite functor spectral sequences

13.1 External spectral sequence operations of Singer . . . . . . .

13.2 Application to composite functor spectral sequences . . . .

13.3 Proofs of Theorems 13.1-13.3 . . . . . . . . . . . . . . . . .

171

. . . . . 172

. . . . . 174

. . . . . 175

14 Calculations of W(n)-cohomology and the BKSS E2-page 183

14.1 When X E W(n) is one-dimensional and n > 1 ................. 183

14.2 A Kiinneth Theorem for W(n)-cohomology ................... 189

14.3 A two-dimensional example in W(2) ....................... 189

14.4 An infinite-dimensional example in W(1) . . . . . . . . . . . . . . . . . . . . . 190

14.5 The Bousfield-Kan E2 -page for a sphere . . . . . . . . . . . . . . . . . . . . . 194

14.6 An alternative Bousfield-Kan Ei-page . . . . . . . . . . . . . . . . . . . . . . 198

15 A May-Koszul spectral sequence for W(0)-cohomology 201

15.1 The quadratic filtration and resulting spectral sequence . . . . . . . . . . . . 201

15.2 A vanishing line on the Bousfield-Kan E2-page . . . . . . . . . . . . . . . . . 202

16 The Bousfield-Kan spectral sequence for Sw"m 205

16.1 Some conjectures on the Ei-level structure . . . . . . . . . . . . . . . . . . . . 205

16.2 The resulting differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

A Cohomology operations for Lie algebras 211

A.1 The partially restricted universal enveloping algebra . . . . . . . . . . . . . . 211

A.2 The proof of Proposition 6.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

A.3 The Chevalley-Eilenberg-May complex . . . . . . . . . . . . . . . . . . . . . . 217

10

Chapter 1

Introduction

The primary object of study in this thesis is the Bousfield-Kan spectral sequence (BKSS)

in the Quillen model category sWom of simplicial non-unital commutative F2-algebras.

This spectral sequence calculates the homotopy groups of the homology completion X^

of X E sWom, with E2 -page given by certain derived functors applied to the Andr6-Quillen

cohomology groups H OX. The approach we take in this thesis is two-fold: we develop an

extensive theory of spectral sequence operations on the BKSS, and we use composite functor

spectral sequences (CFSSs) to calculate the derived functors that form the E2-page.

In 1.1 we recall certain aspects of the theory of the BKSS of a pointed connected

topological space, including a CFSS due to Miller for the computation of its E2-page. There

are a number of useful analogies to be drawn between this classical theory and the content

of this thesis. We give some of the necessary algebraic and topological background for what

follows in 1.2.

In 1.3 discuss the BKSS in sWom in detail, and introduce the unstable spectral sequence

operations referred to in the title of this thesis. There are three types of operations appearing

on the BKSS E2 -page: a commutative product, unstable higher divided power operations

(5-operations), and an unstable action of the (homogeneous) Steenrod algebra. The product

and 6-operations arise as the operations Koszul dual to Goerss' operations on Andr6-Quillen

cohomology, and we define them on E2 using techniques of Goerss and Priddy. The Steenrod

algebra action is essentially that found on the cohomology of a Lie algebra in characteristic

two.

We are able to extend the three types of operations to the higher pages of the BKSS,

and we state in 1.3 how the unstableness conditions evolve from one page to the next.

These changing conditions are explained by the appearance of differentials from certain of

the 6-operations to the Steenrod operations. We also describe how the 6-operations and

product relate to the natural homotopy operations on the target, a relationship clarified by

11

a completeness theorem and a vanishing line theorem, which together show that the BKSS

converges strongly to irX when X C sMom is connected.

We describe the other part of our approach in 1.4-1.6. The derived functors that

form the BKSS E2-page may be analyzed using a sequence of composite functor spectral

sequences. As we explain in 1.4, the algebraically rich BKSS E2-page is the target of the

first CFSS, and a key aspect of our approach is to extend this rich structure into this CFSS,

producing therein a theory of unstable operations. We cannot complete our computations

with this structure alone, so in 1.5 we generalize the construction, forming an infinite regress

of CFSSs, each calculating the E2-page of the last, and each possessing a theory of unstable

spectral sequence operations.

In 1.6, we indicate how it is possible to use this immense amount of structure to make

computations, including a computation of the BKSS E2-page for a sphere in the category

sMom. This special case is important for the computation of the BKSS E 2 page in general,

as it is involved in the description the Ei-page of a May-Koszul spectral sequence which

calculates the BKSS E 2 page for a general connected simplicial algebra.

Finally, in 1.7, we discuss some conjectures which would unify the two parts of our

approach. Were these conjectures verified, we would be able to give a complete description

of the differentials available in the BKSS of a commutative algebra sphere. These conjectures

are supported by the computation we have made of the E2 -page.

1.1. The classical Bousfield-Kan spectral sequence

The homotopy theory s'om has much in common with that of pointed connected topological

spaces, and before we introduce our main results, we briefly recall the analogous classical

theory in this section. The intention of this thesis is to produce an enriched version in the

model category sMom of this classical theory.

Suppose that X is a pointed connected topological space with irX finitely generated

in each degree. The (absolute) Bousfield-Kan spectral sequence of X over F2 is a second

quadrant spectral sequence

[E2 X]I - Ext(H *(X; F2), H*(St ; F2 )) => 7rt 8-X ,

where X is the completion of X at the prime 2. Throughout this thesis, we use the notation

[E2X]' rather than the more standard E2,t for the pages of the spectral sequence.

At least when X is simply connected and wr*X is of finite type, one may view this spectral

sequence as a tool for calculating 7r*X, as 7r*(X ) ' (7rX) determines the 2-torsion in 7rX.

Under certain hypotheses (satisfied for example when X = S' for n > 1) the BKSS admits

12

a vanishing line at E2 [5], and is thus strongly convergent.

The non-abelian derived functors Ext8 are calculated in the category X of non-unital

unstable algebras over the Steenrod algebra (c.f. [52, 1.4]). If we write V+ for the category

of cohomologically graded vector spaces

W = Wn,

the objects of X are graded non-unital F2-algebras W E V+ equipped with an unstable left

action of the Steenrod algebra, i.e. maps:

Sqt : W -+ Wt+i,

p:W, (0 W"' -+- W1+1'

satisfying the usual properties - Adem relations, unstableness relations, and the Cartan

formula. We take a moment to introduce notation, defining the functor of indecomposables

Q X -+ V+ by the formula

W F2_ W/ (im(W 0 W +W) e E> im(W 4 W)) CZ V+.

The BKSS E2-page can be rewritten as the dual left derived functors

[E2 X1, t'= H (7* (X; IF2)) t :=- D ((Ls QX)(7* (X, F2))'),

where we write DV for the linear dual of a vector space V, and insist that D interchanges

homological and cohomological dimensions. We will use notation following this pattern for

the rest of the thesis.

One useful idea is to search for operations which act on the BKSS. Spectral sequence

operations are typically used to produce new elements on the E2 -page and to compute

differentials on those elements. Bousfield and Kan [11, 14] construct a Lie bracket:

ErX]8 ( [ErX] -- [ErX] s+s+'1 for 1 < r < oc,

with the bracket on E, satisfying a Leibniz formula and inducing the bracket on Er,+.

There are two reasons why one might expect such a Lie algebra structure. First, the

commutative operad W and the Lie operad Y are Koszul dual, and even though the theory

of Koszul homology is complicated by the non-zero characteristic, there is an action of 2

on the derived functors calculating E2. Next, there is a graded Lie algebra structure on

13

homotopy groups given by the Whitehead bracket [57]:

[,]I : rrn X 9 7rX -4 7rn+n'-IX ,

and one may ask whether or not this action preserves the filtration, in which case it would

define a Lie algebra structure on Ec. Bousfield and Kan answer this question in the affir-

mative by proving that the bracket at Ec is compatible with the Whitehead bracket, and

they also show that the pairing given at E 2 has the correct homological description.

This appears to be as far as it is possible to pursue this strategy, as at both E2 and E"

we lose hope of finding structure that can be readily described. We do not expect to extract

further structure on E2 using the Steenrod algebra action in X, or at least not any that can

be described so explicitly. The Steenrod algebra A 2 suffers from the inhomogeneity Sq0 = 1.

Were it a homogeneous Koszul algebra (in the sense of [47]), then its Koszul dual would at

very least act on ExtA 2 (IF 2 , M) for an A2-module M, but even this is not the case. There

is no particular reason to think that the situation should be any better for the non-abelian

derived functors defining E2. Moreover, we simply do not understand the natural operations

that exist on 7r,, in enough detail to expect to see uniform structure appearing on E,. After

all, by the Hilton-Milnor Theorem [45, 4], all natural operations on the homotopy groups

of pointed spaces are composites of the Whitehead bracket and unary operations, and a

natural homotopy operation ir.X -+ 7rmX is equivalent to an element of 7rmS'.

Before we break from our extended analogy, we will discuss the considerable task of

calculating the E2-page of this classical BKSS. Performing this calculation is at least as

difficult as the calculation of the E2-page for the classical (stable) Adams spectral sequence,

which appears to be rather difficult. There is, however, the following method due to Miller

[43] for extracting information about the derived functors H . There is a factorization of

QX into

X rii7 QEU 4V

where EU is the algebraic category whose objects are vector spaces V E V+ equipped with

a left action of A 2 such that Sqi : Vn -- V'+' is zero unless 0 < i < n. This modified

unstableness condition is necessary in order that QvO6 satisfies an acyclicity condition, so

that for W E X there is a composite functor spectral sequence

[EcfW]S2,S1 = H (HOm(W)) => H +S2(W)t.

This spectral sequence was an integral part of Miller's proof of the Sullivan conjecture. The

functor H6 0 appearing in the above description is the Andrd-Quillen homology functor on

s#'om.

14

1.2. The various categories sC

In this thesis we will use quite a number of categories of universal algebras, such as the

category Wom of non-unital commutative F2 -algebras, or the category ie of Lie algebras

over F2 . While we introduce certain general notions we will write C for any one of these

categories.

For any category C of universal algebras, the category sC of simplicial objects in C is a

Quillen model category [49]. These model categories have much in common with the category

of topological spaces. For example, an object X C sC possesses homotopy groups 7rX

and homology groups HeX. We use the pragmatic notion of homology that appears in the

spectral sequences that appear in this context, and it does not always coincide with Quillen's

notion of homology derived abelianization. The cohomology groups H*X are defined to be

the linear duals of the homology groups.

In 3, we recall the definition of spheres and Eilenberg-Mac Lane objects in se. These

play the same role in sC as their namesakes in the category of pointed topological spaces,

which is to represent the homotopy and cohomology functors on the homotopy category of se,

respectively. We also present a unified treatment of homotopy and cohomology operations

(and of homology co-operations) for such categories.

In 5 and 6 we present a number of existing examples of homotopy and cohomology

operations in a common framework, constructing cohomology operations using a general-

ization of Goerss' method from 133]. In 6.8 we use this framework to define cohomology

operations for simplicial Lie algebras, and we prove in Appendix A that these operations

coincide with the well-known cohomology operations of Priddy [48].

In 4, we recall Radulescu-Banu's [50] cosimplicial resolution of X E sMom, which we

denote by X E csWom. The resolution X is suitable for the construction of a BKSS for X.

This construction is rather more difficult than that of Bousfield and Kan's F2 -resolution, as

the naYve monadic cobar construction in sC is not homotopically correct. The totalization

of X is the homology completion X^ of X, and the (absolute) BKSS is the spectral sequence

associated with the totalization tower. In 4.1 we perform the homotopical algebra needed

to identify the E1 and E2-pages arising from Radulescu-Banu's resolution.

1.3. The Bousfield-Kan spectral sequence in sWom

At this point we depart from generalities, turning to the homotopy theory of simplicial non-

unital commutative algebras in earnest. We will restrict to the connected objects X C sMom,

which simplifies various aspects of our analysis. As in the classical case, one must know

how the homotopy groups of the homology completion X^ determine those of X. We will

15

demonstrate (Theorem 4.4) that X^ is equivalent to X as long as X is connected. Moreover,

we will prove in Theorem 15.3 that the BKSS admits a vanishing line from E 2 in this case,

and thus strongly converges to the homotopy of X.

As forecast by the discussion in 1.1, it will help to know a little about the natural

operations on the homotopy of simplicial F 2-algebras in advance. Fortunately, we have

the explicit description of homotopy operations which is lacking in the category of pointed

spaces, since they have been completely calculated by Dwyer [26] (and were studied earlier

by Bousfield [8, 6] and Cartan [14]). In summary, 7,X supports operations

6i : wX -+ 7n+iX, defined when 2 < i < n,

A : 7nX 0 7in X -+ 7n+ni

with /t a graded non-unital commutative algebra product, and the 6 i satisfying various

compatibilities which we discuss in detail in 5.4. In fact, these 6-operations satisfy a 6-

Adem relation which is homogeneous, and there is a corresponding unital associative algebra

A. Note that 7r*X is not a left module over the algebra A, because the operations are

not defined in every dimension. This situation can not be remedied simply be defining

the missing operations to be zero, as doing so is incompatible with the Adem relations

on homotopy. Instead, we must adopt language for such situations, saying that A has a

partially defined unstable left action on irX. In general, unstable homotopy operations will

be partially defined, whereas unstable cohomology operations will be everywhere defined

but vanish in certain ranges.

Goerss [33] described the analogue for sWom of the category 'X, and all of the natural

operations on the Andr6-Quillen cohomology H O,,mX of X E sWom are generated by:

P': HomX -4 H n+ilX;

[,] : HmX 0 H X+n'+X;

HmX -+ H1mX.

As we restrict to connected objects of sWom, the operation 3 can be ignored. These opera-

tions satisfy various compatibilities which we recount in detail in 6.6, and we will denote by

W(O) the category whose objects are vector spaces W C V+ equipped with the P2 -operations

and the bracket. The bracket satisfies the Jacobi identity but falls just short of being a Lie

algebra pairing as [X, x] is not always zero. The P' satisfy a P-Adem relation that is ho-

mogeneous. The evident unital associative algebra P acting on H OmX is a homogeneous

Koszul algebra, the P-algebra with Koszul dual the algebra A, and indeed, this is how it was

originally described by Goerss. In 4.1, we identify the E2-page (for connected X E som

16

with 7r.X of finite type) as the non-abelian derived functors

[ E2X1 -_ H O0) (H<*mX)t.

This description begets a laundry list of operations that we expect to see on E2. The

cohomology of a Lie algebra enjoys an action of the commutative operad (the Koszul dual

of the Lie operad). As we are working in positive characteristic, they also support an

action of the homogeneous Steenrod algebra A := EOA 2 , due to Priddy [48]. We will discuss

these operations in detail in 6.8 and Appendix A.1. We construct in Proposition 8.9 the

corresponding natural 'horizontal' operations on E2:

Sq2 : [E2X] -+ [E2% ]+, zero unless min{t, 2} < j s + 1;h t2t+1

y : [E2 ][ ® [E2AXIS' -+ [E2X]

Moreover, we construct in Proposition 8.2 natural vertical operations constituting a (par-

tially defined) action of the Koszul dual A of the P-algebra:

- [E 2 +++1, defined when 2 < i < t.

These operations satisfy various compatibilities (c.f. 8.4, Proposition 8.9 and Proposition

8.2, where we also discuss an extra operation Jv defined when s = 0).

In light of our convergence results for connected X E s'om, the spectral sequence

converges strongly to the homotopy of X:

[E2 X]t ==> r1X,

where we write n := t - s for the topological dimension at [E2 X]t. When considering

how the a-operations and product on 7r.X might relate to the various operations on E2,

one encounters the following issue: if x C [E2X]' is a permanent cycle detecting a class

SE7rX, then (for s > 2) there are more operations 6vx,...,J _x defined on E2 than

there are operations 62T,..., 6,T defined on homotopy. Moreover, the Steenrod operations

at E2 have no counterpart in homotopy.

This situation is quite reminiscent of that described by Dwyer [25], who works in the

spectral sequence of a cosimplicial simplicial coalgebra (such as the Eilenberg-Moore spectral

sequence). In such a spectral sequence, one expects to find Steenrod operations at E2 but

finds too many. Dwyer constructs a-operations and differentials mapping the excess Steenrod

operations to the a operations. In this way, the excess Steenrod operations fail to be defined

at E, and the a-operations become zero by E, an excellent resolution to this problem.

17

Unfortunately, we cannot use Dwyer's operations. Indeed, although the linear dual of a

cosimplicial simplicial coalgebra is a cosimplicial simplicial algebra (of which the resolution

X is an example), the choice of filtration direction is transposed. Instead, we perform

analogous constructions in the dual setting, and describe in 10.6 a theory of operations on

the spectral sequence of a cosimplicial simplicial F2 -algebra, which may be of independent

interest. While defining these operations is a good first step, they are not yet what we

require, as it happens that they can be lifted one filtration higher when we are working in

the BKSS. In 10 and 11, we explain how to construct operations

6Y : [ErX] -+ [ErX]i+i+1, defined when 2 < i < max{n, t - (r - 1)}

Sq' : [ErX1) -+ [ErX]2ji, zero unless min{t, r} < j s + 1

p: [Er X]s 0 [ErX]t -+ [Er-XT]s+'+

with the 6y potentially multi-valued functions, defined when 2 < i < max{i, t - (r -

and single-valued whenever 2 i < min{n + 1, t + 1 - 2(r - 1)}, and the Sqh potentially

multi-valued functions with indeterminacy vanishing by E2r-2, and which equal zero unless

min{t, r} < j < s + 1. All of the functions that are defined on E2 are single-valued, and

indeed, they coincide with the operations defined on He( 0 ), as we show in Proposition 11.2.

Notice the dependence of the unstableness conditions on r, the spectral sequence page

number. It is in this sense that the unstableness conditions evolve as we pass to higher

pages. As r increases, fewer of the 5Y are defined, and the unstableness condition tends to

6 : [EOX]9 -+ [EOX]t+j+ 1, defined when 2 < i < n,

which exactly mimics the unstableness condition on the target 7rX. Moreover, as r increases,

more of the Steenrod operations become zero, and the unstableness condition becomes

Sq3 : [E,,X]s -+ [EOX]s3l, zero unless t < j < s + 1.

In fact, [EQ0 X]s = 0 when t < 2s (this is the vanishing line given in Theorem 15.3) and when

t = 0 (by the connectivity assumption), so that every Steenrod operation vanishes at E,.

This is quite suitable, as there is no Steenrod action on 7rX.

These conditions alone do not explain, for x G [ErX], what happens to the Jyx with

n < i K t - (r - 1) as we pass to higher pages, or what happens to the nonzero Sq X.

This question has a very elegant answer, expressed in the following equation (implied by

18

Corollary 11.6). For n < i < t - (r - 1):

6Y (drx), r < t - i + 1;drt(X) = { rtt-i-I-l

6y(drx) + Sqt + 2 (x), r = t - i + 1.

Generically speaking, this equation states that 6o'(x) remains a cycle until either until x sup-

ports a differential or until page t - i + 1, when it supports a differential hitting Sq t-+ 2 (x).

That is, the Sqv serve to absorb differentials supported by the excess P , and these differen-

tials precisely account for the evolving unstableness conditions.

1.4. The first composite functor spectral sequence

Now that we have a theory of the operations available on the BKSS in sMom, we turn to the

question of calculating it. If we hope to imitate Miller's use of a composite functor spectral

sequence (CFSS), using the factorization

QW(o) - QZ(O) 0 QU(0) (w() (O) v+)

where Z (0) is the category whose objects are graded vector spaces W E V+ which are Lie

algebras under a bracket which shifts gradings,

Wt (D wt _' -+w t+t'+1.

We write 11(0) for the category whose objects are vector spaces V E V+ equipped with an

unstable action of the P-algebra given by operations

P : Vt Vt+i+l

which are zero unless 2 2 *i(W P" W).

As the category Z (0) is not an abelian category, it is a more technical task to form a

CFSS, and we use the method of Blanc and Stover [3]. The key idea in their presentation

is that the derived functors H := L QU(o) take values in the category W(1) of (O)-FI-

algebras, as they are calculated as the homotopy of an object of sz (0). The first CFSS takes

the following form for W E W(0):

[EcfW) 82,81 = H,() (H'(0 W)tS2,81 _=> (H '(0)W)S31+S2

19

We will now unpack this somewhat dense expression, and explain how various unstable

operations defined at the E2-page and the target interact with the spectral sequence.

Objects of the category W(1) are certain bigraded Lie algebras with a certain partially

defined right action of the A-algebra (c.f. 1.5). In 9.2 we calculate the structure of H* W

as an object of W(1) by explicit chain-level computation, after defining in 9.1 an unstable

version of Priddy's Koszul resolution [47] for the functors H*0).

The linear duals HgO)W admit an unstable partially defined left A-algebra action, since

the algebras A and P are Koszul dual, and by Proposition 12.9 there is a commuting diagram

(for 2 : (H* W)

51 (1. 1)edge horn [IW~ $

(H, (O W)8+1 ede om [Ecf W]O'~ > (H (OW)s$l+

In this sense the 6f-operations on the BKSS E2 -page are compatible with the CFSS.

The BKSS E2 -page also supports products and horizontal Steenrod operations, and we

should attempt to identify them in the CFSS. The functor Hw(1 ) may also be viewed as

a Lie algebra cohomology functor, so that we expect horizontal Steenrod operations and

products to appear in [EcfW]2,s1. We use a new definition of these operations that fits

into the framework set out in 6 (deferring to Appendix A the work of showing that these

operations coincide with those constructed by Priddy [48].)

Moreover, just as we expected J' operations on HgO), we expect a 'vertical' left action of

the homogeneous Steenrod algebra on HQ(,), as it is Koszul dual to the A-algebra. Indeed,

we construct in Proposition 8.9 such operations on the derived functors Hw(1 ). Moreover,

Proposition 8.6 applies to Hw(1 ) just as it applies to H,(,), yielding horizontal Steenrod

operations and products, so that ultimately we obtain operations

Sqv : [Ec W 2 -+ [EcfW)2 ,+ -

Sq - : [E2CfW],S2,Sl -+ [EcfW W)2+j'2sl

[E2fW]82,sl 0 [Ef W]P'P1 -+- [EcfW]S2+2+1,l+Pi,

with both the horizontal and vertical Steenrod operations satisfying their own unstableness

conditions.

Now suppose that x C [EcfW) S2,s1 is a permanent cycle detecting an element T E

(H,( 0 )W)S2+S1. The s2+S1-1 operations Sqi, ... , q2 S1+1T are the potentially non-

zero Steenrod operations on . The si- 2 vertical operations SqVx,... , Sqs'x and the 82+1

horizontal operations Sq x,..., Sq82+1x are the potentially non-zero Steenrod operations

20

on x. This is quite reminiscent of Singer's framework [53] (c.f. 13.1), and in 13.2 we use

Singer's methods to extend the operations on [E'fW] to the entire CFSS. The upshot is that

if X E [EcfW]s2s1 is a permanent cycle, then so are all of the above mentioned Sqvx and

Sqhx, and moreover,

Sq'x detects Sq'T (3 < i < si) and Sq'x detects Sq1+i' (1 + j S2 + 1).

That is, the horizontal and vertical Steenrod operations combined detect the horizontal

Steenrod operations on the target. We examine how this plays out for admissible sequences

of Steenrod operations in Theorem 8.15.

We extend the operations on E2 to the entire spectral sequence in two steps. We first

apply external versions of Singer's operations (c.f. 13.1) and compose these operations with

the spectral sequence map induced by a chain-level diagonal map (c.f. 12.3) on the Blanc-

Stover resolution [3] used to define the CFSS. The methods of 12-13 may be useful in

other contexts for defining operations on composite functor spectral sequences.

1.5. Higher composite functor spectral sequences

We have constructed a comprehensive theory of the operations in the first CFSS, but it may

still be the case that the H.) is as difficult to calculate as Hee), which would mean that

the CFSS is of little use for the calculation of H (O). Rather than being discouraged, we

will turn this similarity to our advantage by iterating our approach. In 7 we extend the

constructions summarized in 1.4, defining algebraic categories W(n) and 1(n) for n > 1

such that W(n) is the category of Z (n - 1)-H-algebras and there are factorizations

QW() = Q1 () 0 QUf) (W(n) nu) 1 (n) q V+

There are CFSSs for W E W(n):

[EfW]Sn+2,...,SI= H* ( )W)Sn+2,n+1,...,S =- (H W)Sn+2+n+1,S,..-,S1

and we equip each of these spectral sequences with a theory of operations which generalizes

the theory of the operations SqV, Sq' and [ discussed in 1.4. The key change between the

new CFSSs and that of 1.4 is that the diagram (1.1) changes when n > 1 to a compatibility

for Steenrod operations Koszul dual to the A-algebra action in W(n).

At this point it is useful to summarize the definitions. For n > 1, let U1(n) denote the

21

category whose objects are vector spaces V G V+ equipped with linear right A-operations

(-)Ai : Vss.., - VSn+i,2Sn_1,...,2s1 (1.2)

defined whenever 0 < i < sn+1 and not all of is , .... , si are zero. Let Z (n) denote the

category whose objects are V e V+ equipped with a (typically non-linear) A-operation as in

(1.2) defined whenever i = sn+1 and not all of i, sn, ... , si are zero, which acts as a partial

restriction for a Lie algebra bracket

[, : VS 0,..., I7 8 -S..++s/...,S1+s'

Finally, let W(n) be the category whose objects are simultaneously objects of U(n) and Z (n)

subject to certain compatibilities.

The functor H* W may be calculated by an unstable Koszul resolution, and both

its linear dual H (n)W and the functor H.() W are naturally objects of My(n + 1), the

category whose objects are graded vector spaces M C Vn4+1 with an unstable left action of

the Steenrod algebra, operations

Sq :MSn+,...Si -- M t+1+1,s,+i-1,2Sn-1,...,2s1

which are zero except when 1 < i < sn and i - 1, Sn-1,..., si are not all zero. This

structure is derived in 8.2, using the Koszul duality between the A-algebra and the ho-

mogeneous Steenrod algebra. This differs from the analogous constructions for W(0)- and

U(0)-cohomology, in that HN(0 ) supports one fewer vertical J-operation than HO).

On the other hand, as in the n = 0 case, H,(n) is an example of (partially restricted)

Lie algebra cohomology, so that 'horizontal' Steenrod operations and products appear. In

8.3 we define these operations:

Sqi: (~gX) 2+1,...,31 -+(SnX"+lI+j,2s ,...,.2slS q3, (H* W(fl) nWl(Sl -s ~.

p: (H (n)X)n+1,.,Si (H X)" -( ) n+1+1,sn+P,...,S+P-

so that W(n)-cohomology is also a certain type of unstable algebra over the homogeneous

Steenrod algebra, with the horizontal Steenrod action. We write Mh(n+ 1) for the resulting

category of Vn+1'-graded unstable algebras over the homogeneous Steenrod algebra.

We identify in 8.4 the relations between the My(n+ 1)- and Mh(n+ 1)-operations, which

leads to the definition of an algebraic category Mhv(n + 1) in which W(n)-cohomology takes

values for n > 1.

22

Consider again the CFSS for W E W(n):

[EcfW] Sn+2,...,Si = H-(+) (H(n) W)Sn+2,Sn+,.Si -> (H (, W)n 2+sn 1,Sn,.,S

The target is an object of Mhv(n + 1), while the E2-page is an object of Mhv(n + 2). We

prove in Proposition 12.9 that there is a commuting diagram (analogous to (1.1)) relating

the Mv(n+ 1)-structures on H0Q(n W and Hn W under the edge homomorphism. As in the

n = 0 case, after extending the Mhv(n + 2)-structure on E 2 to the whole spectral sequence,

this structure converges to the Mh(n + 1)-structure on the target.

1.6. Computing with the composite functor spectral sequences

So far, we have not explained how the CFSSs may be used for calculation. First, we make the

following simple observation. Suppose we wish to calculate the group (H )W)fSflS"

for a given choice of indices. The part of the E2 -page that contributes to this particular group

is the following direct sum indexed by pairs of indices s' 2 , s+ such that s'+ 2 + s' 1 =

e (H (f+ (H n(n )W ) n+2S 8+ 'Sn -- S, i)

Now in each summand, either s' 1 = 0 or s'+2 < sn+. Except for the challenges of

understanding the differentials and hidden extensions of algebraic structure, it suffices then

to calculate the groups

(H.N(+k) HY(n+k-1 ) ( - - HgYn)W)n+k+1,.,S'n+i'8,-.-,81

for all k > 1 and for all indices s'+k+1 + -n- + s 1 = +1 satisfying either S'+k+1 = 0 or

Sn+k = 0. It is easy to calculate these groups in either case, as long as we understand the

derived functors

HY(n+k- 1) ... W

as objects of W(nr+k). We undertake these calculations in 9.2. When s'+k+1 = 0 there are

no derived functors being taken, and when s' = 0 the derived functors may be calculated

simply as the cohomology of a (constant, not simplicial) partially restricted Lie algebra.

With this computation in mind we define the Chevalley-Eilenberg-May complex of a

partially restricted Lie algebra in A.3. This complex interpolates between the Chevalley-

Eilenberg complex for the homology of Lie algebras and May's X complex [40] for the

homology of restricted Lie algebras.

This method is employed to prove Theorems 14.4 and 14.6, which together imply Corol-

lary 14.7, that MVh(n + 1) is the category of W(n)-cohomology algebras for n > 1. That is,

23

the Mhv(n + 1)-structure is all of the natural structure on W(n)-cohomology for n > 1.

Finally, we are able to use all of this structure together to calculate, at least as a vector

space, the BKSS E2-page for the commutative algebra T-sphere S"om whenever T > 1. That

is, we calculate the derived functors H, (0)W, where W = HeomS om is a one-dimensional

trivial object concentrated in dimension T > 1.

Finally, we derive in 15.1 a convergent spectral sequence which calculates the E2 -page for

any connected X G s~om of finite type, which we name the May-Koszul spectral sequence.

Its Ei-page may be described in terms of the BKSS E2-pages of the spheres (using Theorem

14.6), and information about the E2-operations in the BKSS for a sphere passes over to

information about the general BKSS E2-page via the May-Koszul spectral sequence.

1.7. The Bousfield-Kan spectral sequence for Swom

In 14.6, we present a small model for the BKSS Ei-page for a commutative algebra sphere.

Given our knowledge of the operations on the BKSS, of the E2-page for SWOm and of the

homotopy groups ir,SWm (c.f. 5.4), a natural goal is the complete computation of the BKSS

for SgOm. In 16.1, we make two conjectures which would together allow us to make this

complete computation. It turns out that E2 is not the right place to start this computation,

and we need to consider classes on E1 and d, differentials in order to see the full picture. The

problem is that certain relations involving the v- and Sqh-operations only hold from E2 .

The conjectures we make would overcome these problems, and would lead to the description

given in 16.2 of the full structure of the BKSS for 56"

1.8. Overview

In 2 we introduce the fundamental notions needed in this thesis: categories of universal alge-

bras and their homotopy, homology and cohomology functors; the Dold-Kan Correspondence

and Dold's theorem; the various types of Lie and commutative algebras in characteristic two;

and the types of spectral sequences we use most extensively.

In 3 we give an introduction to the theory of homotopy operations and cohomology

operations, and discuss II-algebras, cohomology algebras and homology coalgebras.

In 4 we identify the E2-page of the BKSS, and work directly with the Adams tower to

prove a completeness theorem showing that X^ is weakly equivalent to X whenever X is

connected.

In 5-6 we construct a framework in which a number of classically known homotopy

and cohomology operations may be constructed together.

In 7-8 we construct and study a number of homotopy and cohomology operations in

24

preparation for the following chapters, and in 9 we study the unstable Koszul resolutions

related to certain of these operations.

In 10 we perform a generic construction of spectral sequence operations in the spectral

sequence of a cosimplicial simplicial vector space, and explain how this induces operations

in the spectral sequence of an object of csWom. We give a comprehensive study of the

properties of these operations.

In 11, we shift the filtration of the generic operations to define non-trivial operations

on the BKSS, using a construction of Bousfield-Kan and the structure of Radulescu-Banu's

resolution. We catalogue the properties of the BKSS operations in detail.

In 12 we define the infinite sequence of CFSSs described above. In 13 we equip them

with a theory of spectral sequence operations.

In 14 we use the CFSSs to make calculations of the BKSS E2 -page in the most funda-

mental case, that of a connected sphere in sMom.

In 15 we define a May-Koszul spectral sequence which converges to the BKSS E2-page

for any connected X, and describe the May-Koszul El-page using the data of the BKSS

E2 -pages for spheres. As an application, we prove a vanishing line theorem for the BKSS

E2 -page for any connected X.

In 16, using the operations on the BKSS and our knowledge of the E2-page, we conjec-

ture the full structure of the spectral sequence for a sphere in sMom.

25

26

Chapter 2

Background and conventions

2.1. Universal algebras

In this thesis we will be dealing with various categories C of universal graded algebras over

F2, which we will refer to as algebraic categories. The relevant examples include a number of

categories of graded associative algebras, commutative algebras and Lie algebras, categories

of graded unstable modules and unstable algebras. We'll give the background on such

categories of universal algebras in this section.

For us, an algebraic category is a category whose objects are G-graded F2-vector spaces

X = {Xg}geG, for some set G of gradings, equipped with a set of operators of the form

Xg1 x - x Xg, -- + Xk (with n > 1) satisfying a set identities, and whose morphisms are

graded vector space maps preserving this structure. These defining maps will be referred to

as the C-structure maps. This is similar to the definition given in [3, 2.1] of a category of

universal graded algebras.

It need not be true that all of the C-structure maps must be (multi-)linear in a given

presentation of an algebraic category C, but we will always assume that C is monadic over

the category of G-graded F2-vector spaces. That is, the forgetful functor Ue : C -- + V

will admit a left adjoint Fe : V -- + C, and the natural comparison functor from C to the

category of algebras over the monad U2FC on V will be an equivalence.

In our examples, the monad UCFC will admit an augmentation (of monads) E : UeFc a

id, reflecting homogeneity in the relations defining C. This augmentation has the monad

unit r7 : id -- + UeFe as a section, and may be thought of as projection onto generators.

We will generally omit the functor UC from our notation, writing FC as shorthand for

either the monad U2FC on V or the comonad FeUC on C. We will refer to elements of a free

construction FV using notation such as f(vi), thought of as a composite f of C-structure

maps applied to generators vi E V C FC(V). We will say that f(vi) is a e-expression. In

27

this language, the linear maps

FeFV -- +4 FV, V -2 + FOV and FOV c V,

constituting the augmented monad FC on V may be described as follows: p collapses a C-

expression in C-expressions into a single C-expression; q sends a vector v to the C-expression

v; and e projects a C-expression onto those summands to which no (non-trivial) operations

have been applied. For X E C, the comonad structure maps in C,

FCFCX 4 FCX and X +- FCX,

are as follows: on an expression f(xi), A = FC returns the same expression f(xi) in which

the xi E X are viewed as elements of FCX, and p is the evaluation map equivalent to the

C-structure on X.

2.2. The functor QC of indecomposables

Using the augmentation c : F -+ id of monads on V, any V C V becomes an FC-algebra,

i.e. an object of C. We denote this functor KO : V -+ C; it sends V C V to the trivial object

on V, which is V equipped with coaction map the projection e: FeV -+ V. Whenever we

say trivial in this thesis, we will mean having no non-zero operations, and not equal to zero.

In each of our examples, K2 has a left adjoint, QC : C -+ V, which sends X C C to the

quotient of X by the image of its non-trivial operations. The functor QC sends X E C to the

coequalizer in V of p, e : FCX -- X.

Note that FC is a section of QC, since QCFC is adjoint to UCKC = id.

2.3. Quillen's model structure on sC and the bar construction

For any of the algebraic categories C appearing in this thesis we use Quillen's simplicial

model category structure on the category sC of simplicial objects of C [49], [43], [3]. In

this structure, the weak equivalences (fibrations) are the maps which are weak equivalences

(fibrations) of simplicial abelian groups, so that every object is fibrant.

A simplicial object X C sC is almost free if there are subspaces V" g X" for each n > 0

such that the composite FCV -- FCX, -2+ X, is an isomorphism for all n, and such that

the subspaces Vn are preserved by all of the degeneracies and face maps of X except for do.

An almost free object is cofibrant, and every cofibrant object is a retract of an almost free

object [43, 3].

There is a richer notion, that of an almost free map, which is a map X -+ Y in sC

28

such that Y contains a subspace V, for each n such that the V" are preserved by all faces

and degeneracies except for do, and such that the natural map Xn Li FO(Vs) -+ Y" is an

isomorphism for each n. An almost free map is a cofibration, and every cofibration is a

retract of an almost free map.

A cofibrant replacement functor for sC is an endofunctor f of sC equipped with a natural

acyclic fibration c : f =- id such that the image of f consists only of cofibrant objects. One

classical such functor is the standard comonadic simplicial bar construction arising from the

FC d Ue adjunction. As a functor Be : C -+ sC it is defined by iterated application of the

comonad FO to X C C:

BCX = (Fe)s+CX,

with face maps given by di = (FC)ip, and degeneracies by si = (FC)iL. This object is almost

free, with BsOX generated by its subspace V, = (FC)sX, moreover, it is standard [4, 4) that

the augmentation BOX -+ X is an acyclic fibration. This functor may be prolonged to a

functor Be : sQ -+ sse, and by taking the diagonal we obtain an endofunctor Be of se. A

standard spectral sequence argument shows that this endofunctor is a cofibrant replacement

functor.

2.4. Categories of graded F2-vector spaces and linear dualiza-

tion

In this section we introduce notation for the key categories of graded vector spaces. We will

write V for a generic category of graded vector spaces or for the category of ungraded vector

spaces as convenient.

Write r7 for the category of vector spaces with r non-negative homological gradings and

q non-negative cohomological gradings, so that an object V of Vq decomposes as

V Vs,.Sr,...,S1 lt ,...,tI>0

The category V7 is equipped with a tensor product:

(U 0 V)tqe, ','i =/ t9 Ut/''/q . 1 qr -7

s';+s'=si, t'g+tl=tj

We will often discuss maps between graded vector spaces which do not preserve de-

grees. Although we could encode such maps as grading-preserving maps between appropri-

ate suspensions, it will not be helpful to be so systematic. For example, we will often write

V 0 V -+ V for a map which in fact adds one to certain gradings of V, and will avoid

29

confusion by explicitly stating the effect of such a map on degrees.

We will often need to consider the linear dual of a vector space V, and the standard

symbol, V*, will cause ambiguity, due to our already intensive use of superscripts. Instead

we opt for a modifier written prefix, defining the dualization functor D :(7)OP -+ V by:

(DV)8r, ',81 := hom(V q ,. ., '1,F2).

We will shortly define cohomology functors H*X := DHOX, and we will use the position of

the asterisk to indicate which of homology and cohomology we mean. This is not precisely

an exception to our convention, but was worth mentioning.

Often, the vector spaces we are interested in will support an extra grading, the quadratic

grading, so called because certain operations derived from an underlying quadratic operation

tend to double this extra grading. We do not think of the quadratic grading as either

homological or cohomological, so we write it prefix:

V = k> qkV.

We write qVq for the category of objects of Vq equipped with this extra grading.

A common pattern for us will be to consider vector spaces with r non-negative homo-

logical gradings and a single strictly positive cohomological grading:

V= VtSr,...,Si O, t>1

and we write V+ for the category of such objects. For the rest of this chapter we will often

use gradings of this type, simply because they will be used so extensively later in the thesis.

Similarly, there is a category V', and dualization is a functor D : (V+)QP -+ V.

2.5. The Dold-Kan Correspondence

In this thesis we will use each of the following five chain complexes in ch+V- associated

with a simplicial graded vector-space V E sV7:

C' V := V, with differential d = En di;

NnV : o<i<n ker (di : Vn -* V_ 1 ) with differential d = do;

N,V := no<i<n ker (di : V -> Vn_ 1 ) with differential d = dn;

DegnV := o<i<nm(si : Vn_ 1 -- + V) with differential d = En 0 di.

NJV := Vn/DegV with differential d = E nd.

30

There are evident inclusions of N, V and N,- V into C. V, and a projection of C, V onto N. V,

and all of these maps are weak equivalences. Moreover, the composite N'V -+ N, V is an

isomorphism (as is the composite from N-V). It will be helpful to have an explicit formula

for the composite

C" V -+- Nn-V _- NnV

Lemma 2.1. The normalization map

nml = (id + sodi)(id + sid2 ) -- (id + sn-id) : Vn Vn

is an idempotent chain complex endomorphism with image NV and kernel the degenerate

n-simplices of V, so that there is a commuting diagram

NnV>-> CnV

Nn V > >CnV Nn--V

Proof. It is obvious that nml restricts to the identity on NnV, and that (nml-id) has image

consisting of degenerate simplices. By the simplicial identities, for 1 < i < n:

di(id + si-1di) = di + disi-idi = di + id di = 0.

As for 1 j <i, we also have

di(id + sjidj) = di + sji didj = (id + sj- dj)di,

this proves that di o nml = 0 for 1 < i < n, or that nml has image inside NnV. Thus nml is

an idempotent with image NnV. As NAJV -- NnV is as isomorphism, the rest is easy. l

Each NnV retains the internal gradings of V, and the functor N" appears in the cele-

brated Dold-Kan Correspondence [34, 111.2]:

Proposition 2.2 (The Dold-Kan Correspondence). There is an adjoint equivalence of cat-

egories:

N. : sVt ;- ch+V: F,

under which the homotopy groups of V E sV+ (as a simplicial set) are naturally isomorphic

to the homology groups of NV:

(7rnV)',... (HnN* V).

31

A cycle in NnV is an element x C V, such that dix = 0 for 0 < i < n. We write ZNV

for this group of cycles, referring to elements of ZNV as normalized cycles. Note that

ZN;-V = ZNV is the same group of normalized cycles.

For x a cycle in any of the four homotopy equivalent chain complexes calculating 7r.V,

we will write 7 for the equivalence class of x in 7rV.

It will often be helpful to remove the notational distinction between the chain complex

dimension n and the other homological dimensions s, ... , si. That is, we may view 7rV as

a single object of Vf+1 , defined by

(7r*V)ts . . := (7rsr+ V)s .Si

Now for any collection of indices sr+1, .- , si > 0 and t > 1, define:

K ts r+ 1 , S i = F - 0 :F 2 z } 0 : - 0 :<

degrees: Sr+1-1 +r+1 +1 Sr+1 +2

CKts+,r..s F ( .. < -:0 : F2{dh} : F2{h} :E 0 : -..

Here z and h denote are both to lie in internal cohomological grading t and homological

gradings sr, . . . , s1. There is an evident inclusion mn : Kt,. -,+ CKtr+1,sr. For

any V C sV+, we can identify the subspaces of cycles and boundaries with hom-sets:

homsV+(Kt i.,V) 2 (ZNsrV)t ,,...,s, and

homv+ (CKtri.,, V) 2 (NSr+l+V)t5 . Si-

Under these isomorphisms the chain complex differential Nr,++1V -+ ZN5, V corre-

sponds to zn*. In fact, Kt,,. represents 7r*(-)t,,,..., . in the homotopy category of s'V:

in a category of simplicial vector spaces, the distinction between spheres and Eilenberg-Mac

Lane spaces disappears.

A dual theory exists for cosimplicial vector spaces U. We mention the cochain complexes

C"U := U, with differential d = Eli~l d;

NnU := Un/ Eo<i<n im(d' : Un_1 -+ Un) with differential d = do;

NcU := Fo<i<n-I ker (si : Un -+ Un- 1 ) with differential d = d ic.

There are chain complex maps whose composite is an isomorphism:

n"U +-_ CnU <- nU,

32

and an explicit normalization map

nml = (id + ds')- --(id + d2 1)(id + d's0 ) : C"U -+ N2 U

with properties dual to the simplicial version. The cohomology of any of these three equiv-

alent cochain complexes defines the cohomotopy lr*U of U.

Homotopy and cohomotopy correspond under dualization as follows. If V c sV, then

C*DV = DCV, and there is a natural isomorphism lr*DV -+ D7rV given by:

H*C*DV = H*DC.V - DH.C*V, F " (v)"

2.6. Skeletal filtrations of almost free objects

Suppose that X E sC is almost free on generating subspaces V C X. Miller [43, p. 55]

defines a filtration of X by almost free subobjects

0-1 - FOX> >,. F1X: > F2X: : X- colim FmX = X

as follows. For each m, i > 0, write FVi for the subspace of Vi spanned by the degeneracies

of elements of V such that j <; min{m, i}. Then write FmX for the subobject of X which

is almost free on the subobjects FmVi. The inclusions of these subobjects are almost free

maps, and the colimit is evidently X.

Lemma 2.3. For each m > 0, nml(Vm) C Vm, and V, has direct sum decomposition

Vm = (Vm n NmX) D (Vm n DegnX),

natural in maps of almost free objects preserving the chosen almost free subspaces, and such

that V n n NX = im(VM r Vr). Moreover, the map

(Mi si): Vf" -+ Fm- 1Vm

is injective.

Proof. The final statement is implied by [43, Fact 3.9]. That nml preserves Vm is clear from

its defining formula. The direct sum decomposition and the fact about Vm n NmX both

follow from previous observations about the idempotent nml on X, in particular that it

has image NmX and kernel DegmX. L

33

2.7. Dold's Theorem

According to Dold [22] (c.f. [17, Lemma 3.1]):

Theorem 2.4 (Dold's Theorem). Suppose that F : sV+ -+ sV+ is a functor preserving

weak equivalences, for example, the prolongation of an endofunctor of V+. Then there is a

functor Y: V+ -- V+ > such that the following diagram commutes:

rr+1 r+1sV+ F sVh

Moreover, if F is naturally equivalent to a composite F2 o F1 , then Y is naturally isomorphic

to T2 0 T1.

The idea here is that the functor 7r* induces an equivalence between the homotopy category

of sV+ and V+ In fact, the inverse equivalence can be lifted to a functor into sV+, namely

V -- V, \V+V -- sV+

where we view V as a trivial chain complex. Then T can be constructed as TV := wr(F17V).

2.8. Homology and cohomology functors He and H

In this thesis we will always define the C-homology of X E sC by the formula:

H$X := 7r*(QCBCX) = H*N*(QCBCX).

These homology functors are well defined, as the QC -d KC adjunction is a Quillen adjunction

(that K2 preserves fibrations and acyclic fibrations is immediate), and indeed we are free to

use any cofibrant replacement in place of BCX.

It is not always entirely appropriate to call these functors homology. Indeed, Quillen [49,

11.5] defines homology to be the left derived functors of the abelianization functor, and it

is not true in all of our examples that QC models the abelianization functor. Goerss [33, 4]

explains that this does occur when C is the category of non-unital commutative algebras,

but it does not occur when C is the category of restricted Lie algebras [21].

When C is monadic over V+, we may view the groups H2X together as an object of

Vf+1. That is, each homology group HOX retains the gradings of X, and a new homological

grading is added (to the left of the existing homological gradings). We will sometimes avoid

substituting into the asterisk, writing (HX), in place of (H2 X).

34

We define the C-cohomology H*X of X to be D(HX), or equivalently the cohomotopy

groups 7r*D(QeX) of the dual cosimplicial object. As we dualize to obtain cohomology, the

cohomological gradings and homological gradings are swapped, and H*X may be viewed as

an object of V'+1

Lemma 2.5. Suppose that X E sC is almost free with generating subspaces V g Xn. Then

any homology class in HCX 7 rQeX can be represented by the image in QCX" of an

element of Vn n NnX.

Proof. This follows from Lemma 2.3 - simply represent the class in question by an element

of Vn, and then apply the natural map nml. L

This lemma states that we may find representatives for any homology class in the subobject

vn n NnX of Xn, while for other applications it will be preferable simply to pass to the

quotient Vn of Xn. Trivially:

Lemma 2.6. Suppose that X is almost free with generating subspaces V" 9 Xn. Then the

simplicial object {(QCX), } may be identified with the collection of vector spaces {V}, using

the following composite as the zeroth face map of {Vn}:

Vn Xn r- FCVn_1 -E Vn_1,

and using the other structure maps of X, which by assumption preserve the generating sub-

spaces, as the other structure maps of {V}.

2.9. The action of E2 on V0 2

For any vector space V E V, the tensor power V®2 := V ® V has an action of E 2 given by

the map T interchanging the two factors. We will write S2 V for the coinvariants and S2V

for the invariants of this action:

S2V := (V 0 V)E, := (V V)/E 2 ;

S2 V:= (V 0 V)E2 :={x E V 9V Ix = Tx}.

The trace map is the natural linear map

tr := (1+ T) : S2 V -+ S 2 V,

and we write

A 2 V := im(tr) e S2 V/ker (tr)

35

Thus, we may view A 2 V either as a subobject of S 2 V of as the quotient of S2 V by the

subspace generated by elements of the form v 0 v.

For any V E V there is a natural map

S 2DV -- DS 2 V, a 0 _ "v 0 w & a(v)3(w)".

It is an isomorphism when V is finite-dimensional.

Suppose that V and W are F2-vector spaces, and p : S2 V -- + W is a linear map. A

quadratic refinement of p is a function a : V -- W satisfying, for vi, v2 E V and a C F2 :

-(vl + V2) = o(vi) + C(v2) + p(vI 0 V2) and -(avi) = a 2 -(vI).

In fact, the second condition is redundant (over F2 ), and these conditions are equivalent to

the following condition. For any set B, define A 2 B to be the set of subsets of B of cardinality

exactly two. The equivalent condition is that, for every collection of vectors Vb E V and of

coefficients ab E F2 indexed by a set B, in which all but finitely many of the ab are zero,

the following equation holds:

x(abvb) =Za (vb) + S abacp(vb 9vC ).bEB beB {b,c}GA 2B

If f : S2 V -+ W is a linear map, the function v - f(v 0 v) is a quadratic refinement of

tr of, and indeed:

Proposition 2.7. For any linear map p : S2 V -+ W, extensions of p to a linear map

f : S2 V -+ W are in natural bijection with quadratic refinements of p.

Proof. Suppose that V has basis {vb I b E B}. Then S 2 V has basis the set

{tr(vb 9 Vc) I{b, c} E A 2B} U {vb 0 Vb b B}.

This is easy to check for V finite dimensional, and extends to the infinite dimensional case

as S 2 preserves filtered colimits, and we may calculate V as the colimit

colimF 2 (B') = V.B'CB

In particular, an extension f of p is determined by the quadratic refinement v - f(v 0 v).

Thus, as long as we can produce an extension f with -(v) = f(v 0 v) for any quadratic

refinement a of p, we will have the natural construction we need.

36

What remains to prove is that the linear map f defined on this basis by

tr(vb 0 ve) '-- p(vb 0 Vc), Vb 0 Vb + U(vb)

does in fact have the property that f(v 0 v) = a-(v) for all v E V. Indeed, if we write v in

terms of the chosen basis as v = EbeB abVb, then

v o V = ebVb + x aba, tr(vb 0v),bEB {b,c}EA 2B

and we can apply our definition of the linear map f to this expansion directly, obtaining

f (v ov ) : Z= a a(vb) + >3 ab CbP(Vb Vc) = u(v).bEB {b,c}EA 2 B

Corollary 2.8. There is a natural linear map / : S 2 V -+ V, the square root map,

uniquely determined by the requirements:

VIT1 0 v2 +v2 0 v 1 = 0, v'/v v = v for all v1, v2, V E V.

Proof. This map is the unique extension of 0 : S2 V -+ V corresponding to the quadratic

refinement id : V -+ V of 0. 11

The evocative square root symbol is doubly appropriate, as if V is dual to a finite-dimensional

vector space U E V, the linear dual of the square root map,

DV -- + DS 2 V , S2DV

equals the squaring map U -+ S2 U, defined by u i-+ u 0 u.

2.10. Lie algebras in characteristic 2

As we work in characteristic 2, there is more than one available notion of a Lie algebra. An

S(s) -algebra is a vector space L equipped with a bracket L 0 L -+ L satisfying the Jacobi

identity and the (anti)-symmetry condition [x, y] = [y, x]. A Lie algebra (or A(Y)-algebra)

is a vector space L equipped with a bracket L 0 L -+ L satisfying the Jacobi identity

and the alternating condition [x, x] = 0. Finally, a restricted Lie algebra [20, 131 (or F(Y)-

algebra) is a Lie algebra equipped with a squaring or restriction function (-)[ L -+ L,

satisfying the axioms

(x + + +[ [2] + [Xi, x 2 ] and [x 2 , X 2 ] = [X 1 , [xl, X21].

37

The alternating condition implies the (anti)-symmetry condition, and these three types of

Lie algebras form a hierarchy: a restricted Lie algebra is in particular a Lie algebra, and a

Lie algebra is in particular an S(Y)-algebra.

We will write Yie for the category of ungraded Lie algebras, and Yier for the category

of ungraded restricted Lie algebras.

Fresse [31] explains how to construct the monads S(Y), A(Y) and F(Y) on V which

give rise to these structures, starting with the Lie operad Y. For V C V, it is standard that

the functor

S(Y) : V F- (Y(n) 9 V')E,n>1

inherits the structure of a monad from the composition maps of Y. Fresse observes that

the functor

r(Y): V F-4 (Y(n) 0 V@n)Enn>1

may also be equipped with a monad structure, such that the trace map S(Y) -- 17(y) is

a map of monads, and that an intermediate monad may be defined by

A(Y) : V -+ im(tr : S(Y)(V) -4 1F(Y)(V)).

These monads give rise to the three indicated forms of Lie algebras in characteristic 2. Each

of these functors supports a quadratic grading:

qk(P(Y)V) := (Y(k) ( Vk) -k, etc.,

and since Y(2) is one-dimensional, there are natural identifications:

q2(S(Y)V) e S2V, q2 (A(Y)V) - A2 V, and q2 (1(Y)V) , S 2V.

One can identify an S(Y)-algebra with the corresponding map S2 L -+ L, a A(Y)-algebra

with the map A2 L -- + L, and a F(Y)-algebra with the map S 2L -+ L, which is to say, for

instance, that a map S2 V -+ V admits at most one extension to a S(Y)-algebra structure

map S(Y)V -+ V. By pulling back along the natural maps

S 2 V -4 A2V _* S 2V

one can demote a restricted Lie algebra to a Lie algebra, or a Lie algebra to an S(Y)-algebra.

A restrictable ideal in a Lie algebra L is a Lie ideal I of L, equipped with a restriction

38

function (_)[2] : I -+ I, satisfying the following axioms, for X1, x 2 E I and x3 E L:

(xi + x 2 )[2] - X[2] + x[2 + [X1, x 2 ] and [xf2 ,x3] = [Xi, [x, x3]].

In fact, let PRL denote the category of partially restricted Lie algebras, whose objects are

pairs of vector spaces (L+, Lo), equipped with a Lie algebra structure on L+e Lo in which L+

is a restrictable ideal, and whose maps are Lie algebra maps preserving the decomposition

and commuting with the partial restrictions. This category is monadic over V x V, the

category of pairs of vector spaces, and the value of monad FPRL on (V+, V) is just an

appropriately chosen subalgebra of F(Y)(V+ (D Vo). We will refer to homogeneous elements

of L+ as restrictable, and homogeneous elements of LO as non-restrictable.

In 7.1 we will define various categories of graded partially restricted Lie algebras, where

membership of the restrictable ideal is determined by the non-vanishing of certain gradings.

2.11. Non-unital commutative algebras

In this thesis we will work with non-unital commutative algebras except when we specify

otherwise. As for Lie algebras, there are three different notions of non-unital commutative

algebra available in characteristic 2. A commutative algebra (or S(W)-algebra) is a vector

space A equipped with an associative commutative pairing A ® A -- + A. We will work

with these often, and will write Wom for the category of such algebras. In fact, we will so

often discuss simplicial non-unital commutative algebras that we will refer to them simply

as simplicial algebras.

An exterior algebra (or A(W)-algebra) is a commutative algebra A with the property

that x2 = 0 for all x E A. A divided power algebra (or F(W)-algebra) is a commutative

algebra A equipped with divided power operations, as described in [31, 1.2.2] or [33, 21. In

characteristic 2, these operations are all determined by a single operation, the divided square

-Y2 : A -> A, which satisfies

72(xy) = X22(y), Y2 (Ax) = A2 -Y2 (x) and 1/2(X + y) = -Y2(X) + 72(Y) + XY.

Note that the second condition is in fact extraneous over F 2 , and that the last condition

implies that a divided power algebra is exterior. Thus, -Y2 (xy) = x2 Y2(y) = 0.

There is a notion of a divided power ideal of a commutative algebra: an ideal I of a

commutative algebra A equipped with a compatible divided power structure on I. In this

case, I is necessarily exterior, although for x E A and y E I, Y2 (xy) = x272 (y) need not be

zero.

Again, Fresse [31] explains how to construct the monads F "'Or :- S(W), A(%) and

39

F(%) on V which give rise to these structures, using the commutative operad %' instead of

Y2 . Again, there is a quadratic grading definable on these three monads, and each monad is

generated in degree 2, so that a commutative algebra may be thought of as a map S2L -- L,

an exterior algebra as a map A2 L -- + L, and a divided power algebra as a map S 2L -+ L.

The coproduct ALJB in the category of non-unital commutative algebras is the direct sum

A D (A 0 B) G B, with the obvious product. Moreover, the smash coproduct (to be defined

in general in 3.5) is simply A V B := A 0 B. Indeed, coproducts and smash coproducts in

all three of the above categories are given by these formulae.

2.12. First quadrant cohomotopy spectral sequences

Suppose that V,q is a bisimplicial vector space, ungraded for now. We will follow the

standard conventions, those of [53], in defining the cohomotopy spectral sequence of V,

which calculates the cohomotopy of the diagonal IV| of V. For more detail, see [53, 1.15].

There is a double chain complex Cp,qV Cp V = V,q, where we have decorated the

functors Cv and Ch in order to distinguish them from the functor C., being introduced,

and to distinguish the coordinates - we will always refer to p as the horizontal coordinate

and q as the vertical coordinate. The total complex TV, along with one of its two canonical

increasing filtrations, is defined by

n p(TV )n := Ci,n-XV Fp (T V ), := Ci'n-i V.

i=O i=O

The dual total complex DTV admits a decreasing filtration defined by

FPDTV :=ker (DTV -- * DF--1TV).

Correspondingly, H*(DTV) _= r*(DIVI) is equipped with a decreasing filtration. This fil-

tration is evidently finite (eventually stabilizing in any given dimension), exhaustive (having

union H*(DTV)) and Hausdorff (having intersection zero), and one defines

[E07r*(D IVI)]p'q := FP7rP+q(DIVI)/FP+17rp+q (D IV 1).

Then, there is a spectral sequence with

[E2 V]P'q = 7r 7rq(DV)

and differential d, : [ErV]Pq_ -+ [ErV]p+r,q-r+l so that [Er+1V] is the cohomology of the

40

cochain complex ([ErV]; dr), and for each fixed p and q,

[EV]P'q stabilizes to [EV]P'q - [Eo7r*(DIVI)]P'q as r -+ 00.

Typically, V will admit an augmentation to a simplicial object V- 1 C sV, inducing a

weak equivalence JVt -> V_ 1 . All of our augmentations are horizontal maps to a vertical

object, i.e. an augmentation is a simplicial (in q) map:

d8 : V,q -+ V-i,q coequalizing dh, dh : V,q -+ V,q.

In this case, we view the spectral sequence as a tool for the calculation of the cohomotopy

7r*(DV-1), via isomorphisms

[E,,V]p'q - [Eogr*(DV_1)] p'q.

If V is instead a bisimplicial graded vector space V E ssV', then we may regard [ErV]

as an element of Vh+2. That is:

[E'rV]p~q'Sh .i'' := [Er(Vt ,.:,', )]Pq'.

In our application of these conventions we will actually have V E V+, and will write p = Sh+2

and q = sh+1. We will even sometimes have a quadratic grading on V, which will transfer

to a further grading on the spectral sequence, so our spectral sequences will appear in the

format

q [Er V],Sh+2,---,Si := [ r )]os.. S h s+2,8h+l.

2.13. Second quadrant homotopy spectral sequences

Suppose that

0 <1 -- ( 71 +-72+- -.. < 7

is a tower of surjections of chain complexes, with 'T the inverse limit. Then Y, has a

canonical decreasing filtration:

F' = Fm72 := ker (Y, -- + 'mY-)

41

and we define, with the conventional suspensions:

[EO]S := Esker (T' -+ TrS1);

[El]' := H*[Eo]8 = H,_ker (7, -- + 7T-1).

From this data we may derive the following diagram, in which any pair of composable maps

that consists of a monomorphism then an epimorphism is a short exact sequence of chain

complexes:

0 E-O[EO]O E-'[Eo]l E -2 [EO ]2 E-3[EO]3

F1o FO------ F 2 -. F 3

7.F 0 <F :F2: F

0 ____ ___ - - 70--

0 0 E--0[Eo]0 E-1 [E0 ]1 [EO

Taking homology, each short exact sequence of chain complexes creates a long exact se-

quence, and we obtain two exact couples (c.f. [29] or [42, 2.2]), which we juxtapose, using

dotted maps to indicate boundary homomorphisms:

S E-0 H[E1 ]0 E-1 H[E1 ] E- 2 H[E1] 2 H[El]3

H7,, HFO HF1 < HF 2 <HF 3

...H70 HTc, He~ HT H700

0 H7_1 : H7o <H7 HI2

0 0 E~0 H[E1]0 E- 1H[E1 ]' E-2H[E]2

The vertical boundary homomorphisms HTm -- + EHFm+l in fact form a morphism of exact

couples (c.f. [29]), as follows from Verdier's octahedral axiom (in the homotopy category of

chain complexes, c.f. [39, Appendix A.1]) or a diagram chase. Moreover, the two resulting

spectral sequences have the same Ei-page, so that they are identical (c.f. [29, 6]). This

common spectral sequence is simply the spectral sequence of the decreasing filtration F' on

the complex 'T (c.f. [42, 2.2], [7]). The intended target HT has an exhaustive decreasing

filtration, defined in either of two equivalent ways:

F m (H7o) := im(HFm -+ H7o) = ker (HTo -- H7m- 1 ),

42

and one writes [E0 Hoo]s = FsHt_s7c/Fs+lHt_s,>.

One context in which we may make these constructions is when given any sequence of

maps

0 = Ti 1 +-- To +- Ti +- -

in sV. Such a tower may be converted into a homotopy equivalent tower of surjections

S= T_1<+ T' (+-T'<-

and we may perform the above constructions with Wm :CT' . Homotopy equivalent towers

will produce isomorphic spectral sequences from E1 . From this perspective, a straightforward

way to give a map of spectral sequences that shifts filtration is simply to give a map of such

towers with the corresponding shift.

Suppose now that V is an object of (sV)-+, the category of coaugmented cosimplicial

objects in the category of simplicial vector spaces. We think of the cosimplicial direction

as horizontal and the simplicial direction as vertical, so that the coaugmentation of V is a

(horizontal) map from a (vertical) simplicial object V 1 E sV, i.e. a simplicial (in t) map:

do : V-1 -+ Vto equalizing do, : V" -

There is a cochain-chain complex

(CV)t' := ChCtV = V',

with s the horizontal and t the vertical coordinate, whose differential is the sum of the hor-

izontal and vertical differentials. The total complex TV is a chain complex with a canonical

decreasing filtration, defined by

(TV)n = Hts n CVts, d = dh + dv, (Fm TV)n = Ht-s=n CVtss>m

This filtration of TV corresponds to the tower of surjections of chain complexes defined by

(7mV)n := (TV/Fm+1TV)_ Htj-s=n CVts,s<m

which has inverse limit WooV = TV. Again, the two evident filtrations of H,(TV) =

43

H.(W.V) coincide, and the resulting spectral sequences coincide, satisfying

[E0V]: ChCtV, do = d';

[E1V, := Ch7r{V, di = dh;

[E2V]t i7r'tIV.

The differential is of the form dr : [ErV] -+ [ErV]'i+'1, and as ever, [Er+iV] is the

homology of the chain complex ([ErV]; dr). We will work with this spectral sequence in

detail, and will need the following explicit description of the higher pages:

[ZrV1] := {x E (FsTV)t_, |dx E (F+rTV)ts_1};

[ErVs [ZrV s/(d ([Zr+ 1 t

The spectral sequence will sometimes admit a vanishing line of slope a on E2 , i.e. there

will exist a constant c such that:

[E2 V]t = 0 for s > c-+ a(t -s).

In this case, the filtration on H,(TV) is Hausdorff and finite, and for each fixed s and t:

[EV]' stabilizes to [E.V]* ~ [E0H,(TV)]' as r -4 oo.

The coaugmentation induces a map V- 1 ~- Tot V where Tot V is the totalization of

V in the simplicial model category sV [34, VII.5]. Bousfield explains how this relates to the

totalization tower [34, VII.5] of V:

Lemma 2.9 [7, Lemma 2.2]. There are natural chain maps N, Tot, V - TmV for m < oc

which induce an isomorphism of towers r. Totm V -- 4 H,mV. In particular H.(TV) ~

7r* Tot V.

Not only then do we have a tower under 'TOO V ~ C, Tot V, but Tot V accepts the

coaugmentation map from V~ 1 . Of course, the coaugmentation map need not be surjective,

but if we factor it as a composite

Ve> r(V-) ::b Tot V

we may form the following diagram by demanding that the vertical composites be strict

44

fiber sequences

r(V 1 ) Fib0 :--Fib' : <Fib 2 < Fib3

r(V-1 ) r(V- 1 ) r(V- 1 ) r(V~ 1) r(V- 1) ...

0 Tot_ :: Toto :: Toti :: Tot2

and applying the functor C..

We will in general hope that V~ 1 -+ Tot V will be a weak equivalence, and to investigate

whether or not this is so, it will be helpful to be able to identify the fibers Fibm up to

homotopy. For this we recall a useful relationship between cosimplicial objects and cubical

diagrams, explained by Sinha in [54, Theorem 6.5], and expanded on by Munson-Void [44].

We will only present that part of the theory that we need, and refer the reader to [35], [46]

or [44] for the theory of cubical diagrams and their homotopy total fibers. For n > 0 let

[n] = {0, ... , n}, and define T[n] = {S C [n]I} to be the poset category whose morphisms are

the inclusions S C S' so that an (n + 1)-cubical diagram in sV is a functor T[n] -- + sV.

Sinha describes a diagram of inclusions of categories

T -]>T [0] r a. 1[] >T [2]---ho h

The augmented cosimplicial simplicial vector space V A -+ sV may be pulled back

along hm to form an (m + 1)-cubical diagram h* V. After noting that V is Reedy fibrant

(c.f. [10, X.4.9]), Sinha explains that there are natural weak equivalences

Fibm+1 ~ hofib(V- 1 -+ TotV) ~+ hototfib(h*V)

under which the inclusion Fib m +- Fib'+' is identified with the map

hototfib(h* V) -+ hototfib(r*h*V) = hototfib(hm 1V).

As h* 1 V is the 0-cube with value V- 1, the tower of homotopy total fibers is identified up

to homotopy with the tower of the Fib'.

45

46

Chapter 3

Homotopy operations and

cohomology operations

Let C be a category of universal graded algebras, monadic over V+. Our goal is to understand

construct operations on the homotopy and cohomology of an object of se. In 3.2 and 3.3,

we set out dual frameworks in which these operations can be organized, and in 3.10 and

3.5, we describe some useful chain level operations that we will use to construct cohomology

operations in 6.

3.1. The spheres in sC and their mapping cones

Using the forgetful functor Ue : C V+, for any X C sC we may define the homotopy

groups 7rX of X, which we view together as an object of V++1. By the definition of the

model structure on se, the functor 7r, : se -+ V+ 1 is homotopical, which is to say that it

inverts weak equivalences. For any set of indices t > 0 and sr+1,.-. ,si > 0, write:

Se+1 ,.,si: FCKt, .; and

CS := FCCKtr+,.

These are the spheres in se and cones on spheres in se respectively, and we write

sph(C) := { t > 0, s,+1, . .. , s> 0}

for the set of spheres in sC. Note that we were very literal here - the spheres in se are

precisely this set of objects, and not, say, the cofibrant objects in se which are weaklyC't

equivalent to some Sif+1 ,...,Si. For S E sph(C) we write CS for the corresponding cone.

For any S E sph(C), there is an evident cofibration in : S -+ CS. Indeed, for any

47

C'tsphere S = Sfr+i ... ,51, S contains a distinguished normalized cycle, the fundamental cycle:

Z E (ZNSr+1S)t.

and the cone CS contains a distinguished normalized chain, the cone on z:

h E (Nor+1+1CS)r. ' ,

and zn is defined by the requirement that zn(z) doh. For any X C sC, by adjunction:

hom Se(+1,.,SX) (ZNr+1X)t and

home(CSt,...,', X) (Nor+1+1X)tr,...,sI

and indeed zn* plays the same role as above, representing the differential of NX. Moreover,

tCin the homotopy category corresponding to the above model category structure, fSrs,..,i

represents 7rn(-)Sr)...,S1 (c.f. [33, 1] or [3, 3.1.1]) which is why we refer to the objects

.Sr,-,1 as spheres.

3.2. Homotopy groups and C-I-algebras

By virtue of the algebraic structure possessed by X E sC, the homotopy groups 7rX possess

certain natural algebraic structure, that of a C-U-algebra. Indeed, as any given homotopy

group is a representable functor on the homotopy category, natural N-ary operations on

homotopy groups

(7r*X) I x ... x (r*X) t N * (7*+X) 1,.Si (3.1) Sr+J)---S SNr1i. NSr 1..)1

are in bijective correspondence with elements of the group

S ,...,s i N . S)Sr+1,---,Si. (3.2)

Blanc and Stover [3] define a new category of graded universal algebras, the category 7rC of

C-H-algebras, monadic over Vh+ 1, whose objects are graded vector spaces V - Vt+ 1 with a

structure map

4 ~ ... XVN N + 1 ...I (3.3)r+41,...,31 Sr+11 ... S1''

for every such homotopy class, satisfying certain natural compatibilities.

It is a standard formalism to encode these compatibilities as follows. A model [3] in

sC is an almost free object of sC which is weakly equivalent to a coproduct of spheres (for

48

example, Fe6 'V for any V E V+1 viewed as a chain complex with zero differential). A finite

model is a model in which this coproduct is finite. Let H be the V-enriched category with

objects the finite models in se, and morphisms

homr (M, M') : homho(se)(M, M').

Then the category of C-H-algebras may be defined as the category of V-enriched functors

HOP ---+ V that send finite coproducts into products (where by V we mean the category

of ungraded F2 -vector spaces). The category of C-H-algebras is monadic over Vf+1 , with

forgetful functor U 2 defined on a functor A E 7rC by:

(U"CA)' -=A(S2,''

and each of the structure maps (3.3) on UTCA is induced by the corresponding homotopy

class (3.2), viewed as a map in H.

One obtains the free C-H-algebra on a graded vector space V E V7+ 1 using Dold's

Theorem (2.4). That is, one views V as a chain complex in ch+V+ with zero differential, and

applies the Dold-Kan Correspondence and C-free functor, obtaining an object FerV E se,

and then

Fr6 V = 7r*(F6 PV ).

Moreover, as F2 is an augmented monad, so is F'e, via the map

F"'V = 7r.(FoPV) 1*4 -rx(PV) = H*V = V,

and in particular, there is an adjunction Qe -d KTe

The theory above has the upshot that understanding the category irC is equivalent to

calculating the homotopy groups of the finite models. In many cases, this can be performed

by calculating the homotopy of individual spheres, and then using a Hilton-Milnor Theorem

(c.f. 5.5) or Kiinneth Theorem (c.f. Proposition 5.5) to bootstrap up to a calculation on all

finite models.

Lemma 3.1. For any model A in sC, the Hurewicz map 7rA -+ ir.QeA descends to an

isomorphism

y: Q"wrA -- + 7r,QA e H2A.

Proof. A is (cofibrant and) homotopic to a coproduct of spheres, and as such may be taken

to be equal to a coproduct of spheres. As 7r*A is free on generators in correspondence

with the sphere summands, Q"27r*A is simply the vector space with basis their fundamental

classes, which is isomorphic to HeA. E]

49

3.3. Cohomology groups and e-H*-algebras

It will in general be preferable for us to consider algebraic structure on cohomology, rather

than coalgebraic structure on homology: algebra is in general a more familiar subject than

coalgebra, and cohomology has the advantage that it consists of representable functors.

Another advantage is that the theory of cohomology and C-H*-algebras is dual to the theory

of homotopy groups and C-H-algebras, and 3.3 can be (and has been) obtained from 3.2 by

appropriate dualization. On the other hand, using cohomology groups has the disadvantages

associated with double-dualization.

For any set of indices t > 0 and sr+1,- ,si > 0, write:

KC,t .KeKt nK r+1,...i := sr+ ,...,; andCJKC,t :KCC]Kt

.. .+1,1Si Sr,..,

C'tThe K,. 1 ,...,Si are the Eilenberg-Mac Lane objects in sC. In the homotopy category of se,

the object Klsr',,. represents the contravariant functor H.(-),'.Si : se -+ V, c.f. [33,

Proposition 4.3].

By virtue of the algebraic structure possessed by X, the cohomology groups H*X possess

certain natural algebraic structure, that of a C-H*-algebra. As for C-H-algebras, natural N-

ary operations on cohomology groups

(H*X)Sr+1,... 1 x -... x (H*X) >+1,.-. -+ (H*X)Sr+1 ,...,Si (3.4)

are in bijective correspondence with elements of the group

H 1 K 1t 1 x .. x KSiN Sr+1 ... S1 (3.5)Sr+11 ..--,1 Sr+1 .- 1s

The category of C-H*-algebras, monadic over V'+1, has objects graded vector spaces V C

Vr+l with a structure map

S1 1 N Nxr+1 . . . VSr+11-...S1 - + VSr+1 ... S(3.6)

for every such cohomology class, satisfying certain natural compatibilities.

The formalism required to express these compatibilities is as follows. A generalized

Eilenberg-Mac Lane object, or GEM, in sC is an almost free object of se which is weakly

equivalent to a product of Eilenberg-Mac Lane objects K+1 . A finite GEM is a GEM

in which this product is finite. Let K be the V-enriched category with objects the finite

50

GEMs in sC, and morphisms

homK(M, M') := homho(se)(M, M').

Then the category of C-H*-algebras may be defined as the category of V-enriched functors

K -- + V that preserve finite products. The category of e-H*-algebras is monadic over V"+,

with forgetful functor defined on a functor h : K -+ V by:

(UHCh) Sr+1l,--,Sl := h(K 1 ,. ,

and each of the structure maps (3.6) on UHeh is induced by the corresponding cohomology

class (3.5), viewed as a map in K.

One obtains the free C-H*-algebra on a graded vector space V E V"+1 of finite type as

follows. One views DV as a chain complex in ch+V+ with zero differential, and applies the

Dold-Kan Correspondence and Ke, obtaining an object K2PDV E sC. Then:

FH'V - H*KeFDV.

Moreover, FHe is an augmented monad: one applies He to the natural collapse map

FCFV - KeFV, to obtain

KHeV L H*FeFDV <-- HK 6eJDV -: FHeV.

and in particular, there is an adjunction QHC d KHe.

These definitions simplify when we apply them to the dual of a vector space U E V+1

of finite type:

FHC(DU) := gr*DQ~cKeFD 2U i Ir*DQCcKCFU A D7r*QecKCFU

suggesting that the functor FHC is altogether of the wrong variance. It is preferable to work

with the functor

CHe-coalgU := lr*QecKerU

discussed in 3.7.

To dualize a paragraph from 3.2: the theory above has the upshot that understanding

the category HC is equivalent to calculating the cohomology groups of finite GEMs. In

many cases, this can be performed by calculating the cohomology of individual Eilenberg-

Mac Lane objects, and then using a Hilton-Milnor Theorem (c.f. [33, 11] and 6.6) or

Kiinneth Theorem (c.f. Theorems 6.15 and 14.6) to bootstrap up to a calculation on all

51

finite GEMs.

3.4. The reverse Adams spectral sequence

We will now give a description of Miller's reverse Adams spectral sequence [43, 4], which

was used by Goerss [33, Chapter V] to calculate the cohomology of Eilenberg-Mac Lane

objects in s~om.

Suppose that X E sC, and consider the bisimplicial object Qe(B X)q E ssV+. There is

a first quadrant cohomotopy spectral sequence

[E2 Q1B ,X4''"''.'.i = 7rlr q(DQCBCX "'si

converging to H*X := 7rp+qDQCeB0XJ. For each fixed p,

7r* (DQ!BCX) e D7r*(QCBCX)

e DQ7r*(B2X)

SDQ"rCB" X 7r

where the second isomorphism is that of Lemma 3.1, so that

[E2 QCBeX '' '''i ( 7rX ' '..'. I

When C = tom, Goerss equipped the reverse Adams spectral sequence with certain

spectral sequence operations [33, 14], work which can be framed using the external opera-

tions, due to Singer, which we reprise in 13. We discuss Goerss' use of the reverse Adams

spectral sequence further in 12.6, where we compare it to our use of composite functor

spectral sequences.

In this thesis we study the Bousfield-Kan spectral sequence (c.f. 4), which is also known

as the unstable Adams spectral sequence, for the category s%'om. Loosely, we find that

certain of the operations on the BKSS defined in 11 are Koszul dual to the operations

identified by Goerss on the reverse Adams spectral sequence.

3.5. The smash coproduct

For X1 and X2 objects of any algebraic category, for example C, 7re or He (to be defined

shortly), we define the smash coproduct X1 Y X2 to be the kernel of the natural map X1 Li

X2 -+ X 1 x X2. When X1 = X2 = X, X V X has a natural action of E2, and we write

X V2 X for the subobject of invariant elements under this action.

52

When X1 and X2 are objects of sC, taking this strict fiber is in fact homotopically

correct, since the map X1 Li X 2 -+ X1 x X2 is always a fibration, and indeed:

Proposition 3.2. For X1 and X2 in sC, the natural C-H-algebra map

7rr(X1 x X2) -+ 7-X1 x irX2

is an isomorphism. If X1 and X2 are models in sC, the natural C-U-algebra map

7rX1 Li 7r*X 2 -+ 7r*(X1 Li X2 )

is an isomorphism, and there is an isomorphism of short exact sequences:

0 > r*X1 V 7r*X2 > ir*X1 Lu 7rX 2 3r1Xi x r*X2 > 0

0 >F Tr(X1 Y X2) >7r* (X1 uJ X2) >1 7r*(X1 X X2) >0

Proof. The first claim is easy: the forgetful functor is a right adjoint, and 7r, preserves

products (of vector spaces). Consider the commuting diagram

0 > 7rX1 Y 7r*X2 ir*X1 Li 7rX 2 > ir*X1 x 7r*X2 - 0

7r*(X1 Y X 2) > 7r*(X1 LI X2 ) > 7r*(Xi x X2 )

in which the top row is a short exact sequence, and the bottom row is just a three term

excerpt of the homotopy long exact sequence of the fiber sequence defining X1 Y X2. If i

were an isomorphism, the bottom row would also be short exact, and a simple diagram chase

would show that i restricts to the isomorphism we desire.

If X1 and X2 are models, the displayed map i is an isomorphism, since both source and

target represent the free C-I-algebra on generators corresponding to the sphere summands

of X, and X2 taken together. E

3.6. Cofibrant replacement via the small object argument

The homotopy of an object X of sC was defined simply by application of the forgetful functor

U :C -+ V, a definition which is tautologically homotopically correct. On the other hand,

in order to define the homology H2X, as the left Quillen functor Q2 does not preserve all

weak equivalences, we must perform a cofibrant replacement before applying Qe. While the

comonadic bar construction Be described in 2.3 suffices to define the groups H*2X, it lacks

the structure that we will need at various points in this thesis.

53

Radulescu-Banu's innovation [50] was to explain that the cofibrant replacement functor

c : sC -+ sC constructed by Quillen's small object argument [49], which by design already

possesses a natural acyclic fibration e : c -+ id, in fact admits the full structure of a

comonad, with diagonal / : c -- + cc. As explained by Blumberg and Riehl [4, Remark 4.12]:

Proposition 3.3. The endofunctor QecKe of sV admits the structure of a comonad, via

the maps

Q'c'2 e Cc (n)e

CcKB 'G 94 Q~ccK -4 Q cKcQccK2 and Q cKC -(')

where q denotes the unit of the QC H K2 adjunction.

The functor c of the small object argument depends on the choice of sets of generating

cofibrations and acyclic cofibrations. It will be helpful in our applications to include in the

set of generating cofibrations the following important cofibrations:

(1) the inclusion of 0 into any sphere S 1 ,.(2) the cofibration SC U S , -+ Jt,tl defined in 11.5;

(3) the cofibration SC -- + Gt,i defined in 11.7; and

(4) for each cofibration A -+ B just mentioned, the map A' 9 A - A' 0 B formed

using the standard closed simplicial model category structure [49, II.4] on sC.

It will be helpful to have included these maps, because of the following facts about the

small object argument functor cX. It is constructed as the colimit of a (transfinite) sequence

of cofibrations:

0 = cOX > iX :- c2 X >- > c3 X

X

and given an element f : A -4 B of the chosen set of generating cofibrations and a

commuting square

A :, cX

fB X

there is a canonical choice of map B -- cn+1X making

A >N CX

c cn+1X

B :X

commute. Indeed, the map cnX -- + cn+lX is constructed by attaching a copy of B along

54

the image of A in cX, for each such commuting square.

We will use this canonical lift later, and so establish a little notation. There is a function

homsom(St, X) -- + homsom(so, ciX)

denoted a ii , natural in X E sC, and which provides a section of

homsom(St, cX) - k homswom( SO, X).

We define F to be the canonical lift corresponding to the square

0 : cOX := 0

Se C X

Finally, we note that Radulescu-Banu's construction has a convenient (albeit not crucial)

consequence for the construction of homotopy cofibers in sC. Quillen's small object argument

actually provides a functorial factorization

X > : cfac(g) Y

of any map g : X -+ Y in sC, and in this notation, one might say that we have been writing

cX as shorthand for cfac (0 -- + X). There is a commuting square

0- > cY

cX C : cY

which by functoriality induces a commuting diagram (ignoring the dotted map):

0: :- ccY ::- - cYm~j i

cx >-: cfac(cg) ::* cY

Radulescu-Banu's diagonal # is the dotted map in this diagram, and as it is a comonad

diagonal ecy o 3 = idey, so that m o 3 is a section of the acyclic fibration cfac(cg) ~-4 cY.

We may define the homotopy cofiber of g as the pushout

cX >- cfac(cg)

0 > hocof(g)

55

and by virtue of the construction just given:

Proposition 3.4. There is a construction in sC of the homotopy cofiber hocof(g) of a map

g : X -+ Y, implemented by natural maps

cX i4 cY -+ hocof(g).

This is in contrast to the standard situation, where there is at best a natural zig-zag, even

from cY to hocof(g).

3.7. Homology groups and C-H,-coalgebras

There is a commuting diagram

s + QccKe

7 + CeO-coalg +r+1 r+1

in which we are using Dold's Theorem (2.4) to define CHe-coalg, the cofree e-H*-coalgebra

comonad. By Proposition 3.3 and the naturality of Dold's Theorem, this is a comonad

on Vt+ 1. A C-H*-coalgebra is simply a coalgebra over this monad, i.e. any h E V+

equipped with a coaction map h --- Ccoalgh satisfying the standard compatibilities.

The homology HOX of X E sC is a C-H*-coalgebra with coaction map

,r*(QC cX) -* ,r*(Q ccX) -* ,r*(Q cKQCcX) = CHC-coag l(7(QcX)).

If X ~ KeV for some V E sVt, then H2X r CHC-coag( 7r*(V)), and the coaction map of

HCX is none other than the diagonal map of the comonad.

The comparison maps of 3.3 give the dual of a C-H*-algebra of finite type a C-H*-algebra

structure.

Proposition 3.5. If V C V and X, X' E HC-coalg are of finite type, then there are natural

isomorphisms:

DCHC-coalgV r F HCDV;

QHCDX - D PrHC-coalg X;

D(X x X') e DX U DX';

D(X - X') e DX V DX';

where the primitives PrHC-coalg X are defined 3.8, and the smash product X T\ X' in 3.9.

56

3.8. The Hurewicz map, primitives and homology completion

For any X E sC, there is a map 7rX -- + H2X, the Hurewicz map, defined as the composite

7rX 2 r*(cX) -- + r*(QCcX).

Indeed, the Hurewicz map provides a coaugmentation of the comonad CHC-coalg, the natural

transformation a : id -+ CHC-coalg of endofunctors of V+ defined by

V r 'r*(cK'eFV) -+ ir*(QeCKCPV) - F CV.

One reading of this observation is:

Lemma 3.6. If X E sC is in the image of K2, then QeX = UCX, and the Hurewicz map

of X is a section of the composite

HeX := 7r*Q'cX * r*QCX - 7X.

Given that the comonad CHe-coalg has a coaugmentation, we may define the primitives

of a C-H*-coalgebra H as the equalizer (in sV):

HrC-coalg (H ) :H a > CH .coact

We will briefly defer the proof of:

Proposition 3.7. The Hurewicz map 7r*X -+ HCX factors through PrHC-coalg(HeX),

and if X is GEM, the map 7r*X -+ PrHC-coalg(H2X) is an isomorphism. In particular,

for any V E V+ji, Pr HC-coalg(CHe-coalgV) - V.

Radulescu-Banu [50] has constructed a cosimplicial resolution X* of an object X E SC

by GEMs, and defined the homology completion of X to be the totalization X^ := Tot(X*).

This construction is the analogue of Bousfield and Kan's R-completion functor on simpli-

cial sets [121, a construction that has proven extremely useful in classical homotopy theory.

There is an additional difficulty, however, in constructing the cosimplicial resolution X*,

which is not present in the classical context: since not all simplicial algebras are cofibrant,

the naYve cosimplicial resolution (with the coaugmentation drawn dashed)

X- - - --- KCQX : (KCQC) 2X (KeQ)3X

fails to be homotopically correct, and as QCKC - id, fails to hold any interest whatsoever.

57

Radulescu-Banu's innovation was to explain that the cofibrant replacement functor c :

sC -- sC constructed by Quillen's small object argument [49] admits a comonad diagonal

0 : c -- cc (already used in 3.7) and can thus be mixed into the cosimplicial resolution,

making it homotopically correct.

In detail, the diagonal is needed in order to define the coface maps in Radulescu-Banu's

resolution, the coaugmented cosimplicial object

X* : cX - - - - -cK2Q cX c(K"QCC)2X KCQCc)3X

Instead of simply using the unit and counit of the adjunction respectively, one uses the

composites discussed in 3.7:

c + cc + cKCQCc and QCcKo -1K QKC -+ id.

By an application of Dold's Theorem (2.4), if X -+ Y is a weak equivalence, so is

X -+ for each s. Both X and , being group-like, are automatically Reedy fibrant

(cf. [10, X.4.9]), so that the map of completions X^ -+ Y^ is a weak equivalence. This

construction is explained and generalized by Blumberg and Riehl [4, 4].

Comments in [4, 4] show that the coaugmented cosimplicial C-H,-coalgebra HIX* is

weakly equivalent to its coaugmentation HOX as a vector space (c.f. 4.1), which starts to

explain the title homology completion. One says that X is homology complete when the map

cX -+ X^ := Tot(X*) is an equivalence.

In Theorem 4.4 we specialize to the case when C is either the category Wom of ungraded

non-unital commutative algebras the category Yier of ungraded restricted Lie algebras, and

prove that the completion X^ is weakly equivalent to X when X is connected. Analogous

results for topological Quillen homology may be found in [18].

A question analogous to questions studied in [18] and 138] is whether the homotopy

category of connected objects of sQ is equivalent to the homotopy category of cosimplicial

QCcKC-coalgebras. We have not investigated this question.

Proof of Proposition 3.7. The maps do, d' : Xo -+ X1 induce respectively the coaugmen-

tation and coaction maps for 77rX0 = HCX on homotopy, while do : -- + X0 induces

the Hurewicz map. The very existence of this diagram then shows that the Hurewicz map

factors through the primitives. The observation that this cosimplicial object has extra code-

generacies when X = KCV (c.f. [4, 4]) completes the proof. l

58

3.9. The smash product of homology coalgebras

For X 1 , X 2 E HC-coalg connected homology coalgebras, we define the smash product X 1 -

X2 to be the cokernel of the natural map X 1 U X 2 -+ X 1 x X2 .

The theory changes a little in form after passing from homotopy to homology, and in

order to obtain a result analogous to Proposition 3.2, we must introduce the left derived

smash product in se. For A1 and A 2 in sC, the natural map A 1 U A 2 -+ A 1 x A 2 is a

surjection, and so in general very far from a cofibration. We define the left derived smash

product A, -KL A 2 to be the homotopy cofiber of this map. In light of Proposition 3.4, there

are natural maps

c(Ai L A 2) -+c(A1 x A 2 ) -+ A1 7L A 2 ,

and this cofiber sequence induces a homology long exact sequence (c.f. [33, Proposition 4.6]).

The following result and its proof are dual to Proposition 3.2 and its proof.

Proposition 3.8. For X, and X 2 in sC, the natural C-H.-coalgebra map

H (X1 U X 2 ) +-- HeX1 U HeX2

is an isomorphism. If X, and X2 are GEMs in sC, the natural e-H*-coalgebra map

HCX1 x HOX2 +- He(X1 x X 2 )

takes part in an isomorphism of short exact sequences:

0: HeX1 - HeX2 : HCX1 x H"X2 :-HeX 1 L HeX2 : 0

0 : e H(X1 7\L X2) : e H(X1 X X2) : e H(X1 U X2) < 0

3.10. The quadratic part of a C-expression

In this thesis, we will often use a method of constructing cohomology operations used by

Goerss in [33, 5], and here we will set up a framework that can be applied to each case. We

continue to suppose that C is an algebraic category, monadic over V, a category of graded

vector spaces, satisfying the assumptions of 2.1.

For V E V, the diagonal map A : V -+ V E V of V induces a diagonal map F2V -+

Fe(V e V) - (Fe(V))u 2 , and writing i 1 and i2 for the two summand inclusions Fe(V) -

(Fe(V))u 2 , consider the map

(FC(A) + i1 + i2 ) : FCV -+ (F V)u 2.

59

This map factors through (FeV)Y", and is symmetric. We name this factoring the cross

terms:

cr: FCV -+ (FCV) YE2 (FOV),

as it measures the non-linearity in an expression in FeV. We will give an example in each

of the categories 'Wom, Yie and fier, in each case using subscripts to denote membership

of the first or second copy of V:

%'om: cr(vw) = (v 1 + v 2 )(w 1 + W2) + v 1w 1 + v2w 2 = v 1w 2 + w1v 2 ;

Yie : cr([v, w]) = [V1 + v2, Wi + W21 + [vi, Wi] + [v2, W2] = [vi, W2] + [Wi, v 2];

Yier : cr(v[2]) = V[2 1 + V 21 + (V1 + V2)[2] = [VI, V2].

For certain categories of interest to us we will define a decomposition map, natural and

symmetric in X1, X 2 E C:

Je : QO(X1Y X2 ) -4 QC(X 1 ) 0 Qe(X 2 ).

When C = Wom, X 1 V X2 X 1 ® X 2 and Q(X1 Y X 2 ) QX 1 0 QX2 , and we choose

the identity map of this object as decomposition map jom . In other words, the map jworn

is defined by xIX 2 - x 1 0 x2 whenever x 1 E X and x2 C X2.

When C = Yie or C = Yier, we define the decomposition map by

j(n) : [X, - ,Xa ][21 x1 0 x2, if r = 0, a = 2, zi E X1 , z2 E X 2 ,

0, otherwise,

where by [x 1 ,- , Xa] [2r] we mean the r-fold restriction (r = 0 when C = Yie) of some

bracketing of various X1, . .. , Xa from X1 and X2, with at least one Zk must lie in each of X1

and X2 . Any element of the smash coproduct may be written as a sum of such expressions,

so there is at most one map jeom satisfying this equation. That this map is well defined is

less obvious, but nonetheless routine.

Finally, we define the quadratic part map que to be the composite

que : (FeV -s+ (FV) 2 -+ Qe((FeV) VE2 (FOV)) - S 2 (QeFeV) = S2V)

Lemma 3.9. Suppose V E V. Then:

(1) queo0 is the composite F6'mV -- S2V - S2V

(2) quyi, is the composite F"'eV A 2 V tr S 2 V;

(3) quvier is the projection FYie'rV - S 2V.

60

Proof. These are simple observations, and an example is more useful than a proof: consider

the expression e := u + vw + xy 2 C F rnV where u, v, w, x, y are in V. Then

cr(e) v1w 2 + wiv 2 + x1y2 + y~x2 , and

que (e) := jcom(cr (e))

= V 0W22 2= (9 W2 + W1 0 V2 + X1 0 Y2 + Yi 0 X2

V1 0 W2 + W1 0 V2 E S 2(QWomFWOMV).

Parts (2) and (3) are a light modification of a part of Proposition 7.5.

In each category of interest to us, the following equation of maps FeFe V -- + S2 V will

always be satisfied:

que o lpv = que o EFCV + que o Fe

where [t and c stand for the multiplication and augmentation of the augmented monad

UoFC. This is another expression of homogeneity in the relations defining C, which states

that if f(gi) is a C-expression in various C-expressions gi(vij), then

qu(fgi)(vig) = qu(f E(gi))(vij) + E(f)(qu(gi)(vij)).

For an example when C = tom, we specify an expression f(gi, g2, 93) := 9192 +g3 E Fe Fe V

in expressions gi := viivi2 + V3 E FeV for each i = 1, 2, 3. Then

qu(fgi)(vij) = qu((vulvl 2 + v13)(v 21v22 + v 23) + (v 3 1v32 + v33)) = tr(v13 0 V23 + v31 0 v 32),

qu(fc(gj))(vjj) = qu((v1 3 )(v 23 ) + (v 33 )) = tr(v13 0 v 23 ), and

(f) (qu(gi) (vij))= qu(v3 1v32 + V3 3 ) = tr(v 3 1 0 V32).

61

62

Chapter 4

The Bousfield-Kan spectral sequence

In this chapter, we will write C for any category of universal graded F2 -algebras satisfying the

standing assumptions of 2.1. The Bousfield-Kan spectral sequence of X E se is the second

quadrant homotopy spectral sequence (c.f. 2.13) of Radulescu-Banu's resolution X E csC of

X recalled in 3.8. The key objective of this thesis is to understand this spectral sequence

when C = Wom.

Our first step, in 4.1, is to identify the E2-page as appropriate derived functors. Before

we turn to the calculation of these derived functors in later chapters, we consider the con-

vergence target, Tot X =: X^. From 4.2 to the end of this section, we will give a proof of

Theorem 4.4 - that the completion X^ is weakly equivalent to X when e is either Wom or

fier and X is connected.

Although Theorem 4.4 alone does not fully resolve the question of the convergence of the

BKSS, we will prove in 15.2 that if X E sWom is a connected object with H*om of finite

type, the spectral sequence supports a vanishing line at E2 , so that there are no convergence

problems whatsoever when X E sMom is connected.

4.1. Identification of E1 and E2

In light of 3.6 and 3.7, applying the functor He to X yields the monadic cobar resolution

of HOX in the category HC-coalg, obtained by repeated application of the monad on

HC-coalg of the adjunction

UH2-coalg : H1-coalg ; V : C HC-coalg.

63

In more detail, we have a map of coaugmented cosimplicial objects

cX - - - - --- cKCQCcX c(KCQc)2 X

QCcX - - - -- QCK Q~cX QCc(KCQcC)2X --

and if we abbreviate the monad CHe-coalgUHO-coalg on HC-coalg to C, applying 7r,, to this

diagram we obtain a cosimplicial Hurewicz map:

7r*r 7rX - ----- - r--_ < > 7rX X1

Pr(H2X*): Pr(HOX) - -- - Pr(UCHX ) | Pr(C2 HX) ...

HOX* : H2X - - - - - -- CH: C2HX -.

The indicated maps are isomorphisms since each :X for s > 0 is a GEM, thanks to Propo-

sition 3.7. In particular, we see that:

[E1 X]t 2 (prHe-coag (s+1HCX))t;

[E2 X]' f ((R' PrHe-coal)HOX)t.

Corollaries 6.9 and 6.17 and Proposition 3.5 show that

Theorem 4.1. If C is either Wom or 2ie', and X is connected with H*X of finite type,

then HI' is of finite type for each s, and:

[E11] S (C*DQHCBHeH X)s;

[ E2X]i (H HCX .

4.2. The Adams tower

Bousfield and Kan defined the Bousfield-Kan spectral sequence, or unstable Adams spectral

sequence, of a simplicial set in two different ways. Their earlier approach [9] was to define

the derivation of a functor with respect to a ring. This approach constructs the Adams

tower over the simplicial set in question, and lends itself well to connectivity analyzes.

Their latter approach, [12], to give a cosimplicial resolution of a simplicial set by simplicial

R-modules, lends itself more to the analysis of the E2-page, and is directly analogous to

Radulescu-Banu's construction described in 3.8.

Since the release of [9] and [12], the relationship between the two approaches has been

64

clarified by the introduction of cubical homotopy theory [35]. In this section we will define

the Adams tower of a simplicial algebra using a construction analogous to Bousfield and Kan

in [9], and then to relate it to Radulescu-Banu's construction using the theory of cubical

diagrams.

For brevity, write K := KC and Q := QC. For any functor F : sQ -+ sC, we define

the rth derivation RrF of F with respect to homology as follows. The definition is recursive,

and again involves repeated application of the cofibrant replacement functor c:

(RoF)(X) := F(cX),

(RsF)(X) :=hofib((R._1F)(cX) (RF(rc) (Rs_1F)(KQcX)),

where 77 is the unit of the adjunction Q - K, i.e. the natural surjection onto indecomposables,

and hofib is any fixed functorial construction of the homotopy fiber. These functors fit into

a tower via the following composite natural transformations:

( (RsF)(X) -- + (R.- 1 F)(cX) (R-i)(- ) (Rs_1F)(X))

We have thus constructed a tower

S.. (R2F)X > (R 1F)X - (RoF)X = FcX,

which is natural in the object X and the functor F. The functors RF are homotopical as

long as F preserves weak equivalences between cofibrant objects. Employing the shorthand

RsX := (Rsid)X,

we define the Adams tower of X to be the tower

- > R2 X > R1X : ROX = cX.

For example, (R2F) (X) is constructed by the following diagram in which every compos-

able pair of parallel arrows is defined to be a homotopy fiber sequence.

(R2F) (X)

(Ri F) (cX) > FcccX - FcKQccX

(RIF)(KQcX) -- FccKQcX -> FcKQcKQcX

65

In general, (R.+1F)(X) is the homotopy total fiber of an (n + 1)-cubical diagram:

(Rn+1F)(X) := hototfib((R7 +F)X).

See [35], [46] or [44] for the general theory of cubical diagrams. Before defining the cubical

diagram (Ro+1F)X, we set notation: for n > 0 let [n] = {0,..., n}, and define '[n]

{S C [n] } to be the poset category whose morphisms are the inclusions S C S'. Then an

(n + 1)-cube in sC is a functor '[n] -+ sC, and the (n + 1)-cubical diagram (Ro+1F)X:

T[n] -+ sC is the functor defined on objects by:

S -* Fc(KQ)Xnc(KQ)Xn- Ic ... c(KQ)X0cX where Xi := 1 if i E S;

0, if i S,

such that for S C S', the map ((R'+1F)X)(S) -+ ((Ro+1F)X)(S') is given by applying

the counit r : 1 -+ KQ in those locations indexed by S' \ S.

Radulescu-Banu defines the homology completion of X to be the totalization

X^ := Tot(X*) = holim(Totn(X*)),

and the BKSS to be the spectral sequence of the Tot tower

... - Totn(X*) -+ Tot_(X) -+X9

under cX. Our goal in this section is to prove

Proposition 4.2. There is a natural zig-zag of weak equivalences of towers

{ Rn+ 1X} ~ { _- hofib (cX -+ Totn (X*))} .

That is, the Tot tower induces the Adams tower by taking homotopy fibers, and thus the

spectral sequence of the Tot tower coincides with the spectral sequence of the Adams tower.

As X- 1 equals cX, the tower hofib(X 1 -- TotnX*) appearing in 2.13 is one of the

towers in Proposition 4.2. This proposition explains the relevance of the Adams tower to

the cosimplicial resolution, and thus its relevance to the BKSS which was defined as the

spectral sequence of this cosimplicial object.

Proof of Proposition 4.2. By the discussion of the Tot tower in 2.13, it will suffice to con-

struct a weak equivalence h*X* -+ (R'id)(X) of (n+1)-cubes. The (n+1)-cubical diagram

66

h* * is defined on objects by

(h* X*)(S) c(KQc)Xn(KQc)X-l ... (KQc)XOX where Xi:= 1, if i E S,

0, if i V S,

and the map (h* X*)(S) -+ (h* X)(S u {i}), for i V S, may be described as follows. Let j

be the smallest element of S Lu {n + 1} exceeding i, so that

(h* X') (S) c(KQc)X- (KQc)Xa+1((KQ (KQc)xi1 (KQc)xO X, if j < n;

c(KQc)Xi- .. . (KQc)x X, if j = n +-.

In the expression for either case, we have distinguished one of the applications of c with an

underline, and the map to (h* X*)(S Li {i}) is induced by the composite c -- + cc -+ cKQc

of the diagonal of the comonad c with the unit of the monad KQ.

We now define maps (h*X)(S) -+ ((R +1id)X)(S) for S = {jo < ji < ... <jr} G

{0,.... , n}. The only difference between the domain and codomain is that in ((RO+1id)X)(S),

all n+2 applications of c are present, whereas in (h* X*)(S), only r +2 appear. The required

map is then

O'd KQ3inrJr-1 -1 KQ3 r-1 -i2 -1KQ ... KQ3il iOlKQpiOX,

which is to say that we apply the iterated diagonal the appropriate number of times in each

c appearing in the domain. As 3 is coassociative, this definition is unambiguous, and the

resulting maps assemble to a weak equivalence of (n + 1)-cubes. E

4.3. Connectivity estimates and homology completion

In this section we will make the following connectivity estimates in the Adams tower:

Proposition 4.3. Suppose that C is one of the categories Wom or fier, that X E se is

connected, and that t > 1 and q > 2. Then there is some f(q, t) > t such that the map

7rq(Rf(q,t)X) -- + 7q(RtX) is zero.

Propositions 4.2 and 4.3 together imply the following conjecture of Radulescu-Banu:

Theorem 4.4. If either C = 6om or C = Yier and X E sC is connected, then X is naturally

equivalent to its homology completion X^.

Proof of Theorem 4.4. The fiber sequences Rn+iX -* cX -+ TotnX* fit together into a

67

tower of fiber sequences. Taking homotopy limits, one obtains a natural fiber sequence

holim(RnX) -- + cX -- X^.

We need to show that holim(RaX) has zero homotopy groups. Applying [34, Proposition

6.14], there is a short exact sequence

0 -+ lim1irq+i(RnX) -- irq(holim(RnX)) -+ limiF(Rn X) -+ 0.

Proposition 4.3 implies that for each q, the tower {irq(RnX)}n has zero inverse limit and

satisfies the Mittag-Leffler condition (c.f. [10, p. 264]), so that the lim1 groups appearing

also vanish.

The application of the small object argument functor c adds to the difficulty of proving

the connectivity estimates of Proposition 4.3. We circumvent the difficulty of working with

c by shifting to the standard bar construction BC on sC, which we abbreviation to b.

We define recursively a somewhat less homotopical version I'RSF of the derivations RF:

('?oF)(X) F(X),

('Z, F) (X) :=ker (('-T_1F) (bX) (2~1-F)('r7bX)) ('R,-1F)(KQbX)).

There are three differences between this definition and that of RF: here, there is one fewer

cofibrant replacement applied, we use b instead of c, and we take strict fibers, not homotopy

fibers. While these functors are not generally homotopical, we define the modified Adams

tower of X to be the tower

S-. - '"2X Z > 'X >'OX = X,

where JZSX is again shorthand for ('Z8 id)X, and the tower maps 6 are defined as before.

Proposition 4.5. There is a natural zig-zag of weak equivalences of towers between the

Adams tower of X and the modified Adams tower of X. In particular, the modified Adams

tower is homotopical.

Proof. Let CR(sC) be the category of cofibrant replacement functors in sC. That is, an object

of CR(sC) is a pair, (f, e), such that f : sC -- + sC is a functor whose image consists only

of cofibrant objects, and e : f => id is a natural acyclic fibration. Morphisms in CR(sC) are

natural transformations which commute with the augmentations. For any (f, e) c CR(sC)

68

we obtain an alternative definition of the derivations of a functor F : se -+ se:

(R4F)(X) := F(f X), (RfF)(X) := hofib((Rf_ F)(f X) -+ (Rf_1F)(KQf X)).

These functors are natural in f, so that a morphism in CR(sC) induces a weak equivalence

of towers. Our proposed zig-zag of towers is:

Rs = Rcid +- R'Ocid -+ Rid 42- 'ZRb J'Reid = 'Rs

The maps with domain R Cid are induced by the maps cc : b o c -+ c and be : b o c -+ b

and are evidently natural weak equivalences of towers. The map Yo : ('?ob)X -+ (Rbid)X

is the identity of bX, and the map 'R 0c : (Rob)X -+ (Rbid)X is c : bX -> X. Thereafter,

-y. and 'RZe are defined recursively:

('Rs+ 1id)X

('Rs+1b)X

(R$+1 )X :=

ker((Z'Rid)(bX) -+ ('ZRid)(KQbX))

t induced by ('ZseR8 e)ker (('Rb)(bX) -+ ('ZRb)(KQbX))

Sincl.

hofib(('Rob)(bX) -- +- ('Rob)(KQbX))jinduced by (-y,,-y)

hofib((Rbid)(bX) -+ (R'id)(KQbX))

Lemma 4.6 shows that the kernels taken are actually kernels of surjective maps, and by

induction on s, the maps y, and 'Rae are weak equivalences. l

The connectivity result will rely on the observation that any element in the sth level of

the modified tower maps down to an (s + 1)-fold expression in X. In order to formalize this,

when C = Wom, we let PS : sC -+ sC be the "s th power" functor, the prolongation of the

endofunctor Y - Y' of C, where YS = im(mult : Y®s -+ Y). When C = 6om, we define

PS := FS, the sth term in the lower central series filtration (c.f. [13]). Then we have:

Lemma 4.6. Suppose that either C = Vom or C = Yier. The functors 'Zr, 'Rrb and 'RPS

preserve surjective maps and there is a commuting diagram of functors:

- - Rr -2 '1 > R0

> >: Pr+l , ... > p3 > p2 >_ id

Proof. As b and Ps preserve surjections, we need only check the claims about 'RrX for

69

X E se, which is constructed as the subobject

RrX := ker (br-ir7bi : brX - br-iKQbiX)i=1

of bX. For the rest of this proof only we write P as shorthand for F0. In dimension n, this

is the following subset of (brX), := (Fn'+)rXn:

r

(9rX), := ker (p(r-i)(n+1)rpi(n+1) : (Pn+1)rXn -- + (Pn+1)r-iKQ(Fn+1)iXn)i=1

Whichever of %om or Yier we are working with, it is possible to construct monomial

bases for FV once a basis of V has been chosen. For given n and r, first choose a basis of

Xn; build from it a monomial basis of FXn; build from this a monomial basis of F2Xn; etc.

Continue until we have a monomial basis of Fr(n+l)X_ - (brX)n. The effect of the map

p(r-i)(n+1)%pi(n+1) on monomials is either to annihilate them or leave them unchanged,

depending on whether any non-trivial constructions were employed at the ((n + I)i)th stage.

Thus, the subset (JNrX)n has basis those iterated monomials in which some non-trivial

construction was used in the ((n + 1)i)th for 1 < i < r. The image of such a monomial in

Xn lies in pr

To see that 9r preserves surjections: if f : X -+ Y is a surjection, choose a basis

B Li B' of Xn for which f maps the B bijectively onto a basis of Yn and B' maps to zero.

We may continue this pattern at each stage of the construction of iterated monomial bases

of pr(n+l)Xn and pr(n+l)yn. That is, we may choose a basis C Li C' of Fr(n+1)Xn such

that the monomials in C only involve the elements of B and map under f bijectively onto a

basis of pr(n+l)Yn, and such that each monomial in C' involves some element of B', and so

vanishes under f. This pattern is further preserved in passing to the monomial bases just

derived for (RrX), and (RrY)n, proving the claim that JZr preserves surjections. EZ

We are now able to state and prove the key connectivity result in detail:

Lemma 4.7. Suppose that X c sC is connected, t > 1 and s > 2. If C = Wom, then

(gtPs)(X) is (s - t)-connected. If C = yier, then (2tPs)(X) is (log2(s) +-1 - t)-connected.

Proof. We will prove this by induction on t. The induction step is simple: by Lemma 4.6,

there is a short exact sequence:

0 >- (JZP")(X) > (Jt_1P)(bX) - (t1_1Ps)(KQbX) > 0.

Now both bX and KQbX are connected, as they have ro(bX) = irX is zero by assumption,

and 7ro(KQbX) = QirX. By induction we can bound the connectivity of (t_ 1P)(bX)

70

and (3ZtiPS)(KQbX), and the associated long exact sequence shows that (ZtP)(X) has a

connectivity bound at most one degree lower.

For the base case, t = 1, as PS(KQ-) = 0 for s > 2:

(9R1P)X := ker (P(bX) -+ P8 (KQbX)) = PS(bX)

When C = Yier, a modification [13, 4.3] of a theorem of Curtis [19, 5] states that PS(bX)

is log2 (s)-connected. When C = Wom, we must demonstrate then that PS(bX) is (s - 1)-

connected. For this we use a truncation of Quillen's fundamental spectral sequence, as

presented in [33, Theorem 6.2]: the filtration

Ps(bX) D P4-+1(bX) D P,+2 (bX) D

of PS(bX) yields a convergent spectral sequence [EoPs(bX)]q > r(P'(ba)), with:

Nq((Q'ombX) p), if p > s;[ EOP8(bX )]P = ~p

0, if p <s.

As 7ro(QWombX) - QeOm(7robX) = QQOn(0) = 0, the t = 1 result follows from [24, Satz

12.1]: if V is a connected simplicial vector space then V2O is (p - 1)-connected. E

Before we can give the proof of Proposition 4.3, we need the following twisting lemma,

analogous to that of [9]. Before stating it, we note that (JZ 8,3t)X and JZ,+tX are equal by

construction.

Lemma 4.8. The maps 9Zj : &ZX ---+ JZ_ 1X are homotopic for 0 1. Now JZ,+tX is constructed as the subalgebra

s+ts+tX := n ker (bs+t-ibi : bs+tX -+ bs+t-iKQbiX)

i=1

of the iterated bar construction bs+tX, and for 0 < i < s + t, Jikj is the restriction of the

map biebs+ti-1 : bs+tX -+ bs+t'-X. Proposition 4.9 gives an explicit simplicial homotopy

between the maps bebt-1 and bs-lebt. Moreover, the naturality of the construction of

Proposition 4.9 implies that this homotopy does indeed restrict to a homotopy of maps

Rs+tX -- 1JZS+t-IX. E

71

Now that we have the twisting lemma, Proposition 4.3 follows:

Proof of Proposition 4.3. By Proposition 4.5, it is enough to prove that for any q > 0 and

t > 1, 7rq('Zf(q,t)X) -+ rq('ZtX) is zero for some f(q, t) t. Apply Rt- to the diagram

of functors constructed in 4.6 and apply the result to X to obtain a commuting diagram of

functorsTt 6 rkt5 ~

Jkf (q,t)X> t+ Jt

'Rtpf (qxt)-t+X > :ztP 2x > 'tP 1x

By the twisting lemma, 4.8, the composite along the top row is homotopic to the map of

interest, and factors through 'RtPf(qo)-+1X. If we choose f(q, t) = 2t+q-1 when C = Wom

and f(q, t) = 2 t+q-1 + t - 1 when C = Yier, then Lemma 4.7 shows that jRpf(qt)-t+1X is

q-connected.

4.4. Iterated simplicial bar constructions

We will now state and prove a useful result on iterated simplicial bar constructions, used

in the proof of the twisting lemma. The result here applies in general in the category C of

algebras over a monad. Establishing notation, for any simplicial object X in C, we will write

df, : Xq -- + Xq-1 and sfy : Xq - Xq+1

for the ith face and degeneracy maps out of Xq. Suppose that G and G' are endofunctors

of C, that D : G -+ G' is a natural transformation, and that C, C' G C are objects. Write

[,] : home(C, C') --+ home(GC, G'C') for the operator sending m : C -- C' to the

diagonal composite in the commuting square

GC I" > G'CGmI '[4]m IG'm

GC' 41C G'C'

There is an (augmented) simplicial endofunctor, b C s(C2), derived from the unit and counit

of the adjunction:

id - -o,o- - - (Fe)1 Do1 0,0- (Fe)2 (F6)3 .+02,2

The simplicial bar construction b = BC on sC is the diagonal of the bisimplicial object

obtained by levelwise application of b. That is, for X C sC, bX is the simplicial object with

72

(bX)q := (FC)q+lXq, and with

dbx := [Di,q]dq.

The augmentation e : b -+ id is defined on level q by

eq = DOODO,1 -.. DO,q : (Fe)q+1 -+ id.

We can now construct the simplicial homotopy needed for the twisting lemma, 4.8.

Proposition 4.9. The natural transformations fb and be from b2 : Se -+ sQ to b: sC -

sC are naturally simplicially homotopic.

Proof. Write K - b 2 X and L = bX for the source and target of these maps respectively.

Noting the formulae

[Oiq] 2 = [Dq+i,2q 0 Di,2q+1l and [Siq] 2 = [Sq+i+2,2q+2 0 Si,2q+1],

we can describe the simplicial structure maps in K and L as follows:

di ~ -Diq]d-y;[tiiq;

dt' = [ZDq+i,2q 0 ' i,2q+1]djq

Ksiq = [q+i+2,2q+2 0 -6i,2 q+l Siq.-

We can now state an explicit simplicial homotopy between the two maps of interest. Using

precisely the notation of [41, 5], we define hjq : Kq -+ Lq+i, for 0 < j q, by the formula

hjq := [Dj+1,q+2 0 .0 Dj+1,2q+1]Sjq.

We first check that these maps satisfy the defining identities for the notion of simplicial

homotopy, numbered (1)-(5) as in [41, 5]. Each identity can be checked in two parts (a)-

(b):

(1) We must check that diq+hjq=hj _,_id- whenever 0 i <i j q, i.e.:

(a) d isq = sf_ q 1 dy%, and

(b) Di,q+1 j+1,q+2 - Dj+1,2q+1 = Dj,q+1 Dj,2q-1lq+i,2qDi,2q+1-

(2) We must check that d f+1,q+ihq = dj+i,q+ihj+i,q whenever 0 < j q - 1, i.e.:

(a) d+j,+s = dj+,q+1i +1,q, and

(b) Dj+1,q+1Dj+1,q+2 .. Dj+1,2q+1 = Oj+1,q+lbj+2,q+2 ... Dj+2,2q+1-

(3) We must check that djq+ihj,= hj,q_1dK 1 whenever 0 j < i - I < q, i.e.:

(a) df++1fq= 1 d ,q, and

73

(b) Di,q+1Dj+1,q+2 ... Dj+1,2q+1 = Dj+1,q+1 ... Dj+1,2q-1Dq+i-1,2qDi-1,2q+1-

(4) We must check that s[ q ihj,q = h whenever 0 < i < j 5 q, i.e.:

(a) S M1q = Sj+1,q+1Si,q, an

(b) Si,q+1Dj+1,q+2 ... Dj+1,2q+1 = Dj+2,q+3 ... Dj+2,2q+3Sq+i+2,2q+25i,2q+1-

(5) We must check that sf'q +hj,q = h whenever 0 j < i q +1, i.e.:

(a) s q+13' = S' q+18 -1,q, ad

(b) Si,q+1j+1,q+2 ... Dj+1,2q+1 = Dj+1,q+3 .'. Dj+1,2q+3q+i+1,2q+25i-1,2q+1-

Each of these equations follows from the simplicial identities, proving that the hjq form a

homotopy. Finally, we check that this homotopy is indeed a homotopy between the two

maps of interest:

q+ ho,q = [Do,q+1D1,q+2 ... 01,2q+1] (dxq+1q)

= [DO,q+1O,q+2 - DO,2q+l]idXq

is the action of C(bX) in level q, and similarly,

dq+,q+ihq,q = q+,q+10q+l,q+2 -.- Dq+1,2q+1](dx%+,q+1Sqq)

S[q+1,q+1gq+1,q+2 ... Dq+1,2q+1]dxq

is the action of beX in level q.

74

Chapter 5

Constructing homotopy operations

5.1. Higher simplicial Eilenberg-Mac Lane maps

In what follows, we will often have a natural map G whose domain and codomain both

support a switch map T, obtained by interchanging tensor factors. Furthermore, we will

so often use the expression TGT that we introduce the shorthand WG := TGT. Although

this notation is potentially ambiguous, whenever we write OGH, for functions G and H, we

mean (uG)H, not u(GH).

Let {V} be a higher simplicial Eilenberg-Mac Lane map [26, 3], i.e. a collection of

maps

Vk : (CU 0 CV)i+k --+ N(U 0 V)i defined for 0 < k 0, the identity

(1I + )Vk = k + Vk 1 + &Vk-1, if k > 1,

1, if k = 0,

holds on classes of simplicial dimension at least 2k, where:

(1) V : CU ® CV ---> N(U x V) is the Eilenberg-Mac Lane shuffle map, also known as the

Eilenberg-Zilber map, a chain homotopy equivalence inducing the identity in simplicial

dimension zero; and

(2) Ok is the map (CU ® CV)i+k -- + N(U ® V)i which vanishes except on Uk 9 Vk, where

its value is just the projection Uk ® Vk -- + N(U 0 V)k.

Note that as 00 commutes with symmetry isomorphisms, so does V.

75

5.2. External unary homotopy operations

In this section we recall the definition of certain homotopy operations with domain irV for

any V E sV, implicit in [26, 4] (c.f. [8, 6], [14]) and explicit in [33, 3], using the functions

a '-* Vni(a 0 a), NsV -- + N,+i(S2 V).

By postcomposing with the maps S2 V -+ A 2 V - S 2 V, we obtain functions from NnV

to Nn+j(A 2 V) and Nn+i(S 2 V).

Proposition 5.1 [26, Lemma 4.1], [33, 3]. These functions descend to well defined homo-

topy operations:

6 ext :TrV --+ ri(S 2V), defined when 2 < i < n,

Aext :r V -- + ir (A2 V), defined when 1 < i < n,

ext :r V -- + ir i(S2 V), defined when 1 0,

the map -ext .:r V - 7 -r2 n (S 2 V) satisfies

atn(2+ ) = n(V) +ot (1 + T)V(x 9 y) for x, y E ZNnV.

Proof. Although all of the operations are defined in the cited references, we will be a little

more explicit about the definition of -", and the final equation of the proposition.

As described in [33, 3], we might choose to define a-xt using a universal example, for

which the cycle

z 0 z E ZNn(S 2 Kn) - F2

is the only possible representative, demonstrating that the formula -d a 0 a yields the

correct (well defined) operation. To check that o-"t : 7rOV -+ 7roS2 V satisfies the stated

equation, we need only check that it holds on z1 + z2 E ZNo(Ko D Ko) " F2 e F2 . But

aeXt (Zi + Z2) - -eXt (Zi) - 0-t(z2) = zi 0 z2 z 2 z = (1 + T)V(zi 0 z2),

as V is the identity in dimension zero.

To explain the equation when n > 1, as ext (y) (1 + T)Vo(x 0 x), we obtain

e-rt (:T+ a) - 0-*(y) - a-re() (I + T)Vo(1 + T)(x 0 y)

76

and using the symmetry T((1 + T)(x 0 y)) = (1 + T)(x 0 y) and the fact that 0 vanishes

on (1 + T)(x 0 y),

(1 + T)VO(1 + T)(x 0 y) = (1 + wVo)(1 + T)(x o y) = V(1 + T)(x 0 y). U

5.3. External binary homotopy operations

We will now give an account of various natural external homotopy operations, most of

which are binary operations, induced by the Eilenberg-Mac Lane shuffle map V : N.(V) 0

N.(V) -+ N,(V 0 V), which is also known as the Eilenberg-Zilber map. These operations

are well known, but we make a point of giving them the following unified treatment:

Proposition 5.2. There is a natural commuting diagram:

S2 (7r,V) > r* (S2V )proj 7r. (proj)

A 2(7r*V) - r (A 2V)jinci 7r*(incl)

S2 (7 ) >7(2V

For cycles x, y E ZN*(V) and z E ZN (V), the upper horizontal is determined by

T0 g - x 0 y,

and the lower horizontal is determined by

S+ V(x 9 y + y 9 x) and - u (9 e' t(,).

Proof. During this proof, write Vu, VM and VL for the upper, middle and lower horizontal

maps. We must demonstrate: that Vu is well defined; that

ker (7r*(proj) o Vu) ;2 ker (proj),

so that there is a unique map VM for which the upper square commutes; and that one

may extend the composite 7r*(tr) o Vu along the trace map S2 (7r*A) -+ S 2 (7r*A) using the

operations o0 "t. A simple diagram chase would then reveal that the bottom square must

also commute.

As V is a chain map, it produces a well defined map (7r-V)02 -+ 7r*(V02 ), and the fact

that V = wV implies that this map descends to a well defined map VU.

The kernel of the projection S2 (7r*V) -+ A 2 (7rV) is spanned by classes of the form

77

T 0 7, and the image under 7r.(proj) o V of such a class equals x 0 x which is zero as

x O x C A 2 V is zero. This proves the inclusion of kernels.

Finally, to extend the composite 7r.(tr) o Vu to S2(7 rV), we simply need the operations

a-"' to satisfy the equations of Proposition 2.7, which are part of Proposition 5.1. ID

5.4. Homotopy operations for simplicial commutative algebras

Suppose that A E s'om is a simplicial non-unital commutative algebra, with multiplication

map a : S2 A -4 A. Then by composition with the map 7r,(p) : 7r,(S2 A) -+ 7rA, one

obtains unary operations:

7r,(y) o 6 xt : rn A -+ ?rn+iA, defined when 2 < i < n,

and a pairing

A := 7r*(p)o : S2 (7r*A) -+ ,r*A.

Proposition 5.3 [26]. These operations have the following properties:

(1)(2)

(3)

(4)(5)

the pairing p equips 7r*A with the structure of a non-unital commutative algebra;

the ideal Gn>1 IrnA is an exterior algebra;

the ideal En>2 7rnA is a divided power algebra, with divided square given by the top

6-operation, i.e. x - 6nx for x C 7rnA;

the non-top operations, 3i : 7rnA -+ 7rn+iA for 2 < i < n, are linear;

for x E 7rnA, y E 7imA and 2 < i < n

6(xy) 26(X), if m 0;

0, otherwise;

(6) the 6-Adem relations hold: if 6i6jx is defined, and i < 2j, then

(i+j)/3J

E j - ss=[(i+1)/21

A few comments are in order. Firstly, the proposition distinguishes between the top and

non-top 6-operations, as they have rather different behaviour - this will be a recurring

pattern. Secondly, it is not immediately obvious that the 6-Adem relations make sense, in

that it is not obvious that every term in the right hand side is defined. This does indeed

happen, by Lemma 5.4 (to follow).

We may define an associative unital algebra A to be the algebra generated by 6i for

78

i > 2, subject to relations

[(i+j)/3J.s 3(i+:)/=j + 6i+j _-6 , when i < 2j.s- f(i+1)/21

We will say that a sequence I = (ij,...,ii) of integers ij 2 is 6-admissible if ij+1 2ij

for 1 < j < f. For any sequence I = (i,. .. , ii), write 6, for the composite 6j, -6i

This 6-Adem relation allows us to write any 6 _ in A as a sum of composites 6 j in which

J is 6-admissible. In fact, it follows from [33, Proposition 2.7] that the algebra A has an

admissible basis, consisting of those 61 = 6i, . - 6i, with I a 6-admissible sequence.

It then makes sense to make the following definition. Suppose that I is any non-empty

sequence of integers at least 2, and J is a sequence of integers no less than two. Then we will

say that I produces J in A, denoted I - J, if 6j appears with non-zero coefficient when 61

is written in the 6-admissible basis of A. In this case, J must be 6-admissible and I must

be 6-inadmissible unless J = I.

Proposition 5.3 does not state that 7rA is a left module over A, since the 6-operations

are not always defined (or even linear). We define

m(I) := max{(ii), (i2 - fi), (i3 - i2 - fi), ... ,(ie - - f)},

following the convention that max(0) =-o, for any sequence I of integers ij > 2 (or more

generally, for any sequence of non-negative integers). The intent of this definition is that

the composite 6J, by which we mean

7rnA r+iA - - 4 7r+i+-+ie A,

is defined if and only if n > m(I). Note that when I is a non-empty 6-admissible sequence,

m(I) = if - if-1 - -- - - ii =: e(I),

the Serre excess of I. Moreover, if I is 6-admissible, then for any expression 6 i, - - - 6i 1x there

is some k with 0 < k < f such that each of the k operations 6i, - . -6ft-k+1 are acting as top

operations, and each of the remaining f - k are acting as non-top operations.

The following lemma shows that the 6-Adem relations make sense as they appear in (6).

ALemma 5.4. If I -+ J, then m(I) m(J).

Proof. It is enough to show this result when I and J are distinct and have length two, in

light of the evident algorithm for expressing 61 in terms of admissible composites. In the

length two case it can be checked directly from the format of the 6-Adem relation, and the

79

inequality is in fact strict (unless I is itself 6-admissible).

Finally, one should note that these operations generate all of the operations in the

category 7r'om, and that all of the relations between the operations in 7rfom are implied

by those presented here. Goerss [33, 2] presents this information as follows. First, he

observes that there is a Kinneth Theorem available:

Proposition 5.5. Suppose that A 1 and A 2 are models in sMom. Then 7r.(A1 WA 2 ), which is

the coproduct of 7r.A1 and 7rA2 in 7rom, may be calculated as the non-unital commutative

algebra coproduct of rTA1 and 7rA2 .

After giving the calculation on a single sphere, the homotopy of finite models (which is

the structure defining the category 7rWom) will be determined by this proposition, and the

calculation for a single sphere is the following:

Proposition 5.6 [33, Proposition 2.7]. For n > 0, let In be the fundamental class in

,rn(Swom). There are isomorphisms of non-unital commutative algebras:

7r*(Sworn) S(e)[to] = FWOM~z]

7r*(S 'Om) e A(e)[ 1 (IN) I is 6-admissible, e(I) n] for n > 1;

7r* (Swom) F (e)[3r(I,) I I is 6-admissible, e(I) < n] for n > 2.

5.5. Homotopy operations for simplicial Lie algebras

Suppose that L E sYie is a simplicial Lie algebra with bracket {,] : A 2L -* L. There are

unary operations

Ai :=r-(*, ]) o Aext : 7rnL -> rn+iL, defined when 1 < i < n,

which we write on the right as x '-4 xAj, and a bracket

[, ] := 7r*([, ]) o t : A2 (ir*L) -- 7r*L.

Alternatively, one can suppose that L G sYier is a simplicial restricted Lie algebra with

bracket [,] : S 2 L -+ L, and construct operations:

Ai : r([]) o o t : 7rnL -- + rn+iL, defined when 0 < i < n, and

Pro,[20, ] :Fr( L)o : S2(rL) s o rL.

Proposition 5.7 [13], [20, 8]. For L C sfie, these operations satisfy:

80

(1) the bracket gives -r.L the structure of a Lie algebra;

(2) the ideal En,1 Ir7L is a restricted Lie algebra, with restriction given by the top A-

operation, i.e. X[ 2 1 = Anx for x E 9rnL;

(3) the non-top operations, Ai : rL -+ 7n+iL for 1 2j, then

xAA 2 = (i-2j)/2-1 i- 2j - 2 - k)A A

k=O

For L E s~ier, we may omit (1), modify (2) to state that the whole of 7r.L is restricted, and

modify (3)-(5) to include Ao.

Similar comments apply as for commutative algebras, for example, one needs Lemma

5.8 (to follow) to understand why this unstable relation makes sense.

The well known A-algebra is the unital associative algebra generated by A for i > 0,

subject to relations

AjA (i-2j)/2-1 (2j - 2 - k) A ~j1kA2j+1+k for i > 2j.k=0

We say that a sequence I =(i, ... , ii) of non-negative integers is A-admissible if ij+l 2ij

for 1 < j < f. For any sequence I = (ii, ... , ii), if we write Al for the element Ai, Ai,

in A, then the A-algebra has the evident admissible basis, and we may make sense of the

symbol I A- J. Note that the ordering of the generators in AI is opposite the ordering for

the 6 I, to be consistent with the fact that we write the A-operations on 7rL on the right.

Thus, we may think of A, as the composite operator

7r, L -- * r+iL -4 ... -: 4 7r+i,+.-+,L,

again defined only when m(I) n, so that 7r.L is not a right module over A. We will say,

however, that it is an unstable partial right A-module. Note that when I is a non-empty

A-admissible sequence, m(I) = ii, not the Serre excess, reflecting the observation that when

xAi1 - - -Ai, is a A-admissible composite, the top (i.e. restriction) operations which appear

are applied first. The following lemma shows that the A-Adem relations make sense in (5).

81

ALemma 5.8. If I + J, then m(I) m(J), and J does not contain zero unless I does.

These operations generate all of the operations in each of the categories 7rtie and 7r.Yie',

and the relations presented here are sufficient, as:

Proposition 5.9 ([20, Theorem 8.8 and proof], [13]). For V c V1 , choose a homogeneous

basis of V, and construct from it a monomial basis B of A(Y)V (such as any choice of Hall

basis). Then:

Fl b E Bt, I A-admissible with m(I) < t, adI does not contain 0

F 7rierV F Ab b E Bt, I A-admissible with m(I) < t,I does not contain 0 when t = 0

F2 \I (b [2r b E Bt, I A-admissible with m(I) < t,r { 0

For the sake of interest, we can emulate Goerss' method of calculating the cohomology

of GEMs in seom (c.f. 6.6, [32] and [33, 11]) by giving a Hilton-Milnor decomposition

for the calculation of the free Yie-H-algebra on a finite-dimensional object of V 1 , using

[51, Proposition 3.1]. For any i > 0, write EZF 2 E V, for a one-dimensional vector space

concentrated in homological dimension i. For any finite collection of indices i, ... ,in > 0,

we would like to calculate:

F re(E11F2 E ... ( EtF2) = 7r*Fie(Ki, e - -E Kin).

and we obtain a decomposition of 7r*Fy.(Kil D ... E KiQ) as follows. For any monomial b is

the free Lie algebra on {X1,... , xn} and any collection of n vector spaces A 1,..., An, there

is a corresponding tensor product wb(A1,. . . , An). For example, one defines

W [[X2 ,XiX 3] := A 2 0 A 1 o A 3.

Moreover, for each monomial b there is an evident function

Wb(A1,. ... , An) -+ F ie (A, E .. -- E An),

given in our example by a2 0 al 0 a3 -+ [[a2, ai], a3]-

Iteration of the procedure described in [45, 4.3], using the formula of [51, Proposition

3.1], we obtain a Hall basis B of the free Lie algebra on {xl, ... , Xn}, with the property that

the resulting map

® Fy"Wb(A1, ... , An) -+ Frie(A1 E ... e An)bEB

82

is an isomorphism, natural in A,, ... ,An. Thus, there is an isomorphism in sVi:

DFrie Wb(Kil,.. Kin) -+- F-' e(Kij (a .. --D Kin).beB

Moreover, if we follow [33, 11] by writing jk(b) for the number of appearances of Xk in the

monomial b, there is a homotopy equivalence

Wb(Kil, ... ,Kin) - KE-= abi

Thus, on homotopy there is a decomposition:

F"ieE(E=1k(b)ik)F2 F~r e(Ez'F2 .. , ebEB

under which the fundamental class of a summand on the left maps to the corresponding Lie

bracket of fundamental classes on the right. This proves the first part of Proposition 5.9.

83

84

Chapter 6

Constructing cohomology operations

6.1. Higher cosimplicial Alexander-Whitney maps

Let {Dk} be a special cosimplicial Alexander-Whitney map [56, Proposition 5.2], i.e. maps

Dk : (CR 0 CS)i+k C(R S) for i, k > 0,

natural in cosimplicial vector spaces R, S, with the properties:

(1) dDk + Dkd = (1 + w)Dk-l for k > 1;

(2) D0 is a chain homotopy equivalence inducing the identity in dimension zero;

(3) the restriction of Dk to CR 0 CjS is zero unless i > k and j > k; and

(4) Dk maps CkR 0 CkS identically onto Ck(R ® S).

It is a natural convention to define Dk - 0 for all k, i E Z, in which case the relation

dDk + Dkd = (1 + w)Dk-1 holds for any k.

Maps dual to these are described in detail, under the name special cup-k product, by

Singer in [53, Definitions 1.91 and 1.94], and were developed originally in [23]. Indeed, we

will use these maps later, and denote them

(Dk)* : C(U 0 V)i -- (CU 0 CV)i+k for i, k > 0,

natural in U, V E sV. The sense in which these maps are dual to the Dk is captured in the

following commuting diagram (for i, k > 0):

(CDU 9 CDV)i+k D k C(DU 0 DV)i

(D(CU 0 CV))i+k (DC(U 0 V))'

85

This is the first instance of a notational convention we will use occasionally in what follows.

We consider the operations Dk to be 'of primary interest' in this thesis, and so we prefer

not to adorn the symbol Dk. However, we would like to have access to operations (Dk)* of

which the Dk are the duals in the sense of the above commuting square, and not the other

way around. So, we use a star as opposed to an asterisk when writing (Dk)*.

6.2. External unary cohomotopy operations

In this section we recall the definition of certain cohomotopy operations with domain 7r*U

for any U C cV, using the functions

a - D'-(a 0 a) + D'-+1(a 0 da), C'U -+ Cn+'(S2 U).

The same arguments as in [53, 1.12] show that

Proposition 6.1. These functions descend to well defined linear operations:

qixtrU -+ 7rnrk(S2U), zero unless 0 < k < n.

If U = DV for some V G sV, then we may use the natural transformation S2 D -- + DS 2

to form the following composite, also denoted Sqext:

ir"DV sqt qn+kS2DV -_,rn+k D S2V.

This will be part of the process we use shortly to define cohomology operations.

6.3. Linearly dual homotopy operations

Whenever V E sV has 7rV of finite type, the linear maps Sqixt r"DV - rnkDS2 V

induce dual operators

ir*(S2 V) -+ rkV.

Following [33, 3], one can do much better than just this observation, giving a direct defini-

tion of such operations, valid for any V C sV, whose duals are the Sqext.

Again, the cohomotopy operation Sqext is of primary interest, and we prefer to allow it

its standard symbol (albeit with the attached subscript). On the other hand, we are about

to produce a homotopy operation of which it is the dual, so we will use a star and not an

asterisk. That is, for any V E sV, there is an operation

(Sk* r(S2 V)

86

such that the following diagram commutes:

7rn+kDS2v ((Sqt)*) 7rnDV

'3t7rn+kS2DV Sq

In order to define these precursor homotopy operations, Goerss [33, Proposition 3.7]

observes that any element of x E 7rm(S2 V) can be written as a sum

x =Z r,(1 + T)(yj 9 zj) + J Uk (Wk),O<k [m/2J

where Wk c 7rm-kV for 0 < k < [m/2], and yj, zj E 7r.V, and that we may define:

(Sq k t)*(X) := Wk.Wk

One might only need to determine the operations (Sqext)* for k < m/2, so that when

m is even we may ignore the dual of the top operation, (Sq, x 2 )*. In this case, it is more

convenient to rewrite the key equation as:

x = V(v) + 5 Uk(wk),

Ok<m/2

where v C (S2 (7r*V)), and Wk C 7rmkV for 0 < k < m/2.

6.4. External binary cohomotopy operations

Again, the arguments of [53, 1.12] imply:

Proposition 6.2. Suppose that U E cV. Then there is a pairing

i-text : S2 (7r*U) -+ 7r*(S2U),

defined by x 0 y H-+ D0 (x 0 y), with the property that p-ext(a 0 a) = Sqk ta for a E 7rkU.

Unsurprisingly, these bear relation to the homotopy operation V : 7r.S2 V --+ S2 r*V,

via a commuting diagram, for any V E sV:

DwrS2V DS2hrpV

t xt7r*S2DV -dE S2,1*DV

We might have denoted V by pe* t, but decided against the idea.

87

6.5. Chain level structure for cohomology operations

We will now give generalizations of Goerss' constructions in [33, 5] which yield useful

structure on the complexes calculating C-cohomology in a number of cases. Suppose that

X c sC is almost free, with V g X, the freely generating subspace. Then for each s, the

functor

Home(Xs, -) Homv(V, Ue-)

is naturally an F2-vector space. Writing os = Fe(A) : X, -+ X, Li X,, the addition

operation on Home(X,, -) is given by (f, g) - (f Li g) o p. Now let le be the sum of

((do U do)os) and (ypsido) in the F2-vector space Home(X,, X,_ 1 U X,_ 1 ). It is completely

formal to check that Ze maps to zero in the group

Home(X,, X,_1 x X,- 1) = Home (Xs, X,_1) x Home (X,, X,_1),

and thus le factors through a unique map ce : X, -+ X,- 1 Y X,_ 1 . Furthermore, e enjoys

the symmetry -re = e, and it is again formal to verify the analogue of [33, Lemma 5.5]:

Lemma 6.3. When the equation que o ipv = que o eVe V + que o FCcv of 3.10 is satisfied,

the map Qce induces a chain map of degree -1 on normalized complexes:

NS(QCX) -+4 NS_1((Q'2(X Y X))E2)

The composite

(Ns(QCX) Qe N_ 1((Qe(X Y XE2) + NS_1 (S2(QCX))

is essentially que, in that if v E V, n N8 X represents an element of NSQCX, writing dov =

f(wj) for wj E V-1, we have Oc(v) = que(f)(wj) E S2 y

The typical use of this structure is to define cohomology operations using the external

cohomotopy operations defined above, i.e. natural operations on H*X = 7r*(D(QCBCX))

defined by the composites:

Hn, X 0 Hn2X e"4 xn1+n2D(s2(QeB0X)) - H +n2+X,

He Xs nkD(S2(2BCX)) He1s

e h 7ds nr D(S2(QeBex)) -d e apsX.

These operations are the duals of natural homology co-operations, defined using the maps

88

of 6.3 and 5.3:

(2 HCX), V- r (S2 (QeBCX)) - He+ X,

H,(X) ( r) (+k((eBeX)) H Hn+k+1 X)

Instead of proving Lemma 6.3, we will prove the more general:

Proposition 6.4. Suppose that 0 : FCV -+ GV is a natural transformation from Fe to

another endofunctor G of V satisfying the condition:

0 0 AV = 00 EFeV + 0o Foev : FeFeV -4 GV.

Write 0 : QCXs - G(QOX,_ 1 ) for the following

QCXs -2+ V +o* FeV_ 1 -- + GV- 1 -*- G(Qe X, 1 ).

Then do o O = 0 o (do + d 1 ), and dj o 0 = 0 o dj+ 1 for j > 1, so that 0 restricts to a degree

-1 chain map on NsQCX -- + Ns_ 1 QeX and also on CSQOX -+ C8 _1QeX.

Note that 6 depends on the almost free structure chosen.

Proof. In order to see that d3 o W = G o d.+1 for j > 1, we examine the diagram

V d FeVs_ 1 GV_ 1 GQOXsldj+1 Id j Id GQe(dj)

Y do 0 Y eVS_ 1 FeVs_ 2 > GV_ 2 C GQCXs-2

The dotted vertical arrows are available since X is almost free. That the left square com-

mutes is a simplicial identity, and the center square commutes by naturality of 0. In order

to show that do o 0 = W o (do + di), we use the following diagram, which commutes except

for the leftmost square:

VSd FeVs 1 0 GVs-1 GQOXs-ld1 +codo Fe (codo) G(eodo) IGQC(do)

._1 > FV- 2 > GVs- 2 - > GQOXs - 2

To show that the outer rectangle commutes, it is enough to see that the two composites

V - FeV- 2 are coequalized by 0. Using the simplicial identity dod1 = dodo, we are

trying to show that 0dodo + doedo and OFO(edo)do are the same map from V, to GVs- 2.

Even more, we will show that 0do + 0dOE and OFO(cdo) are the same map FeVs- 1 to GV 8- 2.

89

Starting with an expression f(vi) in various vi C V. 1 , we calculate Odof (vi) = O(f dovi),

Odoef(vi) = O(E(f)(dovi)) and OF(edo)(f(vi)) = 9(f)(e(dovi)). That these three terms add

to zero was the requirement specified for 0. E

6.6. Cohomology operations for simplicial commutative alge-

bras

Goerss [33, 5] defines cohomology operations, natural in A c swom:

Pi= *o o Sqi : HOomA -- + Hj+ +1 A; and

[ = *om 0 Ipext H m A 0 H o, A -+ H+MM+1 A.

He also defines a natural operation /3 HOmA --- HelomA. Note that as a result of the use

of 0*om, these operations have a grading shift.

Proposition 6.5 [33, 5]. These operations have the following properties:

(1) the bracket gives H OmA the structure of an S(Y)-algebra (with grading shift);

(2) the operation / acts as a restriction defined only in dimension zero, so that for x, y E

Hm A and z E H omA:

(x + y) = /(x) + /3(y) = [x, y], and [/(x), z] = [x, [x, z]];

(3) the self-bracket operation on H Om A equals the top P-operation:

Pnx = [x,x] for x E H omA;

(4) if x E H omA, then P'x = 0 unless 2 0,

[x, Piy] = 0;

(7) the P-Adem relations hold: if i > 2j, then

PiP'x = i+j-2 (2s i - 1 pi+j-spss=i--j+

In this case, (7) does state that H OmA is a left module over 'P, the Steenrod algebra for

commutative algebras over F2 of P-algebra. This is the unital associative algebra generated

90

by symbols P' for i > 0, modulo the two sided ideal generated by PO, P1 , and the evident

P-Adem relations.

A sequence I = (ir,.. , ii) of integers ij > 2 is P-admissible if ij+1 < 2ij for 1 < j <.

For any sequence I = (i, ... , ii), write P1 for the monomial P't ... P"1 in '. It follows from

[33, Theorem I] that 'P has an admissible basis, consisting of those P' = P't ... P"' with I a

P-admissible sequence. Again, we will say that I produces J in T, denoted I -+ J if, when

PI is written in the P-admissible basis of ', Pj appears with non-zero coefficient. In this

case, unless J = I, J is P-admissible and I is P-inadmissible.

We define

Th(I) := max{(ii), (i 2 - il - 1), (i 3 - i 2 - il - 2), ... , (i - - i - + 1)},

for a rather different purpose than in 5.4 and 5.5: although the composite

P, : (HOmA i Hn+il+lA H

is always defined, it is forced to be zero (by (4) alone) except when n > Yif(I).

As in 5.4 and 5.5, if a non-empty sequence I is P-admissible, we can identify which

term is largest in the maximum defining T7(I), and calculate that 7j-(I) = ii. More explicitly,

in a non-vanishing admissible expression Pie . . . P"'x, for x E HomA, the only P-operations

that can be a top operation is P"1 .

The following result shows that whenever an expression P'x is forced to be zero by (4)

and we reduce Pax to a sum of P-admissible composites, then (4) forces all of the resulting

summands to be zero.

Lemma 6.6. If I - J, then 47(J) > T(I), with strict inequality when I and J are distinct

and of length two.

The main theorem of Goerss' memoir is that these operations generate all of the opera-

tions in the category HWom, and that all the relations between them are implied by those

presented here. In [33, Chapter V], Goerss shows that the listed operations completely cap-

ture the cohomology of an object KwO". He proves a Hilton-Milnor Theorem [32], which he

uses in [33, 11] to bootstrap up to a calculation of the cohomology of any GEM in swom,

namely [33, Theorem I]. The result states that whenever V E V, is a vector space of finite

type, not only is F'OWV generated by V under the operations of Proposition 6.5, it is as

large as is conceivable given the relations presented. We will present, in Proposition 6.8, a

partial version of his result.

It is interesting to observe that 9', the Steenrod algebra for commutative algebras, is in

fact Koszul dual (c.f. [47]) to A, the algebra which possesses an unstable partial left action

91

on the homotopy of a simplicial algebra. Indeed, Goerss calculates T as the Koszul dual of

A, using a reverse Adams spectral sequence due to Miller [43] (c.f. 3.4). We will explore

this duality further when we consider the Bousfield-Kan spectral sequence.

6.7. The categories W(O) and tL(O)

Suppose that A C sWom is connected, i.e. that iroA = 0. Then HOmA - Q""mrom A = 0,

so that the operation 3 can be ignored. This is a convenient by-product of working with

the cohomology of connected objects, although the real reason that we do so is that doing

so avoid completion and convergence problems. If we say that V C V1 is connected when

V0 = 0, we may identify the full subcategory of V1 on the connected objects with V+.

Goerss' result proves that the monad FH'om on V1 preserves V+ (and indeed it is a

general fact that no non-trivial natural cohomology operations decrease dimension). We

will write W(0) for the category of connected %om-H*-algebras, so that the monad Fw(o)

is simply the restriction of FHWom to V+. The way that we will report Goerss' result here

is to explain how the monad Fw(o) may be constructed on objects of V+ of finite type.

Let the category of unstable T-modules, denoted U(0), be the category whose objects are

V+-graded T-modules in which P acts with grading i + 1 by everywhere defined maps

pi : V - Vn+i+l

which equal zero unless 2 n}

= ('o V) /F 2 {PI O v V Vn, 7f(I) > n, I is P-admissible}.

Proof. This follows from Goerss' [33, Theorem I] and Lemma 6.6. El

Now an object of W(0) is in particular an object of U(0). It is also a (degree shifted)

S(2)-algebra. Thus, there is a natural map

FUO)FS(Y)V -- + Fw(o)V.

This map is not an isomorphism, but it follows from [33, Theorem I] that it is surjective.

Moreover, our final reading of Goerss' result is:

92

Proposition 6.8. For V E V+ of finite type, FW(O)V c V+ is the coequalizer:

coeq ') 9 Fs(2) sbi 0' Fs(Y)V FU(O)Fs(y)Vsb2

where the maps sb1 and sb 2 are defined on 20 (FS(Y)V)m by

sbi(P' Ox) = P' 0 [x, x] and sb2(P' O - PPm O X

Choose a homogeneous basis of V, construct from it a monomial basis of A(Y)V (such as

any choice of Hall basis), and then lift these monomials in the evident way to a collection B

of elements of S(6)V. Then a basis of Fw(O)V is

{P'b | b E B, T(I) < |bl, I is P-admissible} .

Corollary 6.9. Suppose that V c V+ is of finite type. Then so is FW(O)V.

The following observations will be useful for the calculation of the cohomology of objects

of W(O).

Lemma 6.10. The monads Fu(0) and Fw(o) on V+ may be promoted to monads on the

category qV+, by insisting that quadratic gradings add when taking brackets and double when

applying P-operations.

It is typical to think of V E V+ as an object of qV+ concentrated in quadratic grading one

when considering Fw(o)V.

An object of W(O) is in particular an object of U(O), and (as all of the P-operations are

linear), we can define a functor QU() : W(O) V V+ which takes the quotient by the image

of the P-operations. Moreover:

Lemma 6.11. For X z W(O), the vector space Qu(o)X E V+ inherits a (grading shifted)

Lie algebra structure from the bracket of X, yielding a factorization:

QW(O) = QA(Y) 0 Qu(o): (W(O) -+ A(Y) -- + V+).

Moreover the composite QU(o) o Fw(0) equals the free construction F2 0C).

Proof. One checks that the bracket is well defined in the quotient, and that taking the

quotient by the top P-operation imposes the relation [x, x] = 0, to create a A(Y)-algebra

from the pre-existing S(2)-algebra structure. The final claim follows from Proposition

6.8. D

93

6.8. Cohomology operations for simplicial Lie algebras

A standard definition of the cohomology of a simplicial Lie algebra L E sfie or syier is

presented in [48] as follows. Let UL be the simplicial primitively Hopf algebra obtained

by applying the universal enveloping algebra functor levelwise, or the restricted universal

enveloping algebra functor when working in sSier. Applying the Eilenberg-Mac Lane sus-

pension functor ([48, 2.3], [43, 5] or [41, p. 87]), one defines (using a subscript W to avoid

confusion):

H L : 7r*DWUL, if * > 0;

0, if * = 0.

We discuss universal enveloping algebra functors in Appendix A.1. The suspension W

destroys the associative algebra structure but leaves a simplicial cocommutative coalgebra

structure on WUL, with diagonal we denote by A. Homotopy operations for simplicial

cocommutative coalgebras are well known, being the mode of definition of the cup product

and Steenrod operations present in the category X discussed in 1.1 of unstable algebras

over the Steenrod algebra, and can be constructed using Propositions 6.1 and 6.2:

Sqk := ao Sq: (ir"D(1UL) dSq n+kDS2(WUL) * rn+kD(WUL));

A:= A* 0 Pext : (S2 (7r*D(WUL))n !-n 7rnDS2 (WUL) A r4 D(WUL)).

The operations here make H* L a module over the homogeneous Steenrod algebra discussed

in 1.3, which is the usual mod 2 Steenrod algebra 'with Sq0 set to zero'. That is, the

homogeneous Steenrod algebra is the unital associative algebra A generated by symbols Sq

for j > 1, subject to the homogeneous Sq-Adem relation:

[i/2J.

Sq'Sq = S (-k2 -)Sqi+j-k Sqk for i < 2j.k=1

We only ever work with the homogeneous Steenrod algebra and the homogeneous Sq-Adem

relation, and so may omit the word homogeneous if we desire.

This algebra is Koszul dual to the opposite of the A-algebra (c.f. 5.5). There is an index

shift in this duality, so that Sqi corresponds to Ai_ 1 for i > 1 [47, 7.1].

In [48], Priddy concentrates on simplicial restricted Lie algebras L, and works out all of

the natural operations on H* and the relations between them. Moreover, he gives a spectral

sequence argument showing that the two notions of cohomology are isomorphic, with a shift

in degree arising from the use of W: HaL H7L'-L for n > 1.

For our purposes it is better to work in the framework set out in 6.5, giving an alternative

94

definition of Priddy's operations. This alternative definition will fit more readily into the

spectral sequence arguments we intend to make.

For now, let C stand either for Yie' or Yie. Our definition of the cohomology operations

is:

k-1 0*

Sqk :=* o Sq , : H L Sq 7n+k-1D(S2 (QeBeL)) - Hn+kL

At := M I ext : (S2H*L)n "4 r"nD(S2(QeBeL)) H+ Hen1L.

We will check the properties of these operations using a spectral sequence argument similar

to Priddy's, although we will need to give a richer construction of the spectral sequence in

order to extract information about the operations. This work will be deferred until Appendix

A, and will prove:

Proposition 6.12. There are commuting diagrams:

0* oSq~Jx' 0i ________

~*n2k Hnl~n2 1Hgn L Hen+k L Hn 1 L (9 HAnI L HBn1C 2+L

H+ 1 L s* sq Hn+k+1L H1+1L 0 H 2+lL H11+n2+2LW W W W W

That is, the two definitions of Sqk coincide, as do the two definitions of p.

Given the use of suspension W, one expects the notion of top Steenrod operation to be

different to that in other settings, and in this context we say that Sq+ 1 : H'L --+ H 2n+1L

is the top operation.

Proposition 6.13 [48, 5.31. These operations have the following properties:

(1) the product p gives H, L the structure of a commutative algebra (with grading shift);

(2) the squaring operation on H*L equals the top Steenrod operation:

Sqn+ 1 x = X2 for x E H L;

(3) if x E H 'L, then Sqix = 0 unless 1 0,

Sq'(xy) = 1-_i'(Sqkx)(Sqi-ky);

(6) the homogeneous Sq-Adem relations hold, making H*L a left A-module.

This fact follows from Proposition A.3. We will also use the following calculation:

95

Proposition 6.14. If C = Yie (as opposed to Yier), then Sq1 = 0. In particular, for

x HieX, x2 = 0.

Proof. It is enough to prove this for the universal example z, E H_ 2 ieKn . The reverse

Adams spectral sequence ( 3.4) is of the form

E"' = ( K Ye)q ==>y Hp+qKie2 7ieO,n) A-Lie K

The point now is that K'ZY, which is just a constant object in s(7wYie) with value a one-0,n

dimensional Lie algebra in internal dimension n, is actually free as an object of 7r.ie below

internal dimension n + 1. This is simply because there is no A 0 operation defined in wrYie.

One may thus construct pas-A-pas a simplicial resolution of KJ'Ye (a process described in

[1]) which in positive simplicial dimension is concentrated in internal dimension at least

n + 1, implying that Ep'q = 0 when p 1 and q < n. Moreover, E20'q = 0 unless q = n,

showing that H'lIKye = 0. This group contains Sqzn. ElFie n

A sequence I = (it, . .. , ii) of integers ij 1 is Sq-admissible if ij+1 > 2ij for 1 < j < 1.

For any sequence I = (it,... , i1 ), write Sq' for the monomial Sq' - Sql'. The homogeneous

Steenrod algebra has the expected admissible basis, and we say that I produces J in A,

denoted I 4 J if SqJ appears in the Sq-admissible expansion of Sq'.

We use the function m defined in 5.4, this time noting that the composite

Sq : (H omA S-' H + A -- H + A)

is forced to be zero by (3) alone except when n > m(I) - 1.

If a non-empty sequence I is Sq-admissible, we have

m(1) = e(I) = it - it_1 - . . - - ii,

the Serre excess of I. We now have enough notation available to describe the category

HIier, using Priddy's calculations. The results are similar to those in 5.4 on the category

7rom. There is again a Kiinneth Theorem:

Proposition 6.15. Suppose that K1 and K2 are finite GEMs in slier. Then Hier K1 and

HierK2 in H9ier are of finite type, and their coproduct H_,,er(K1 x K2 ) in H-fier may

be calculated as the non-unital (grading shifted) commutative algebra coproduct of HjerKi

and HgerK2 .

96

Proof. We rely on Proposition 6.12 and the following calculation:

H (K1 x K2 ) := 7r*DWU(K1 x K2 )

L gr*DW(UK1 0 UK 2 )

C 7r*D(WUK1 0 WUK2)

e D(,r*WUK1 0 7r*WUK2)

D H*K1 0 H*K2 .

This containment is in fact an equality when H*K1 and H* K2 are both of finite type, in

which case H* (K1 x K2 ) is also of finite type, and the isomorphism is proved. Thus, by

induction on the total number of factors KVier of K1 and K2 , we only need to check that

H*Kl9 er is of finite type for any n > 0. This is implied by a calculation of Priddy [48, 6.1]

which we recall in Proposition 6.16.

After giving the calculation on a single Eilenberg-Mac Lane object, the cohomology of finite

GEMs, and thus the category Hzier is determined by Proposition 6.15 and the Cartan

formula. The structure defining 7rYier is then well understood in light of:

Proposition 6.16 148, 6.1]. For n > 0, let z be the fundamental class in H ner(K9 ie

Then, as non-unital (degree shifted) commutative algebras:

H2ier(K,,e) e S(W)[Sq 12 I I is Sq-admissible, e(I) < n].

Corollary 6.17. Suppose that V c V+ is of finite type. Then so is FH~ierV. That is, the

restriction of the monad FH~ierV on V1 to V+ preserves objects of finite type.

The case of simplicial Lie algebras mimics that of simplicial commutative algebras: for

Lie algebras, the homogeneous Steenrod algebra acts on cohomology, and is Koszul dual to

the opposite of the A-algebra, which possesses an unstable partial left action on homotopy.

Further material on the cohomology of Lie algebras is deferred to Appendix A.

97

98

Chapter 7

Homotopy operations for partially

restricted Lie algebras

7.1. The categories Z(n) of partially restricted Lie algebras

For each n ;> 0, we will be interested in certain categories of Lie algebras monadic over Vn,with a grading shift. Broadly, a Vt-graded Lie algebra is a graded vector space L E V+

with a structure map A 2 L -- L which shifts gradings as follows

L9 L'.-+ L s ,.....+..S1 S1 Sn+S'".'S1+S'

If we wished to be precise we could view the Lie operad as an operad in (V+, ®) such that

and then a V-graded Lie algebra would be an algebra over the corresponding monad A(2)

on V+. In our context, the Lie operad arises as the Koszul dual of the commutative operad,

through the constructions in [33, 5], and the use of the operadic bar construction (c.f.

[30, 3]) explains the shift. See [30, 5.3.4] for a discussion of Koszul duality of operads in

positive characteristic. From this point forward we will simply think of such a Lie algebra

as a vector space L E V+ with a map A 2L -- + L shifting degrees as described.

A Vt-graded partially restricted Lie algebra is to be a V7-graded Lie algebra such that

certain graded parts admit a restriction operation. Specifically, there is to be defined a

restriction operation

(--) : Lt -+LT 2 1wa . a.W i de" te2sa. t y s b s )

whenever not all Of Sn,. ,81 are zero. We will denote the category of such objects Z (n).

99

It is monadic over Vf, with an adjunction

F'C(') : V+ ;: (n) : Un).

The monad of this adjunction may be constructed as an appropriately chosen submonad

of I(Y) : V+ -+ V+ (with 2 shifted as above), containing A(Y). As such, the free

construction FzM")V admits a quadratic grading as in 2.10, which we denote qkFz(n)V.

7.2. Homotopy operations for s1 (n)

We will now state precisely how much of the structure given in 5.5 carries over to our new

setting. If L E s12(n), we may restrict the structure map [,] : Ffn)L -+ L to a map

[,] : q2F2 () L -+ L, with A 2L C E-q2F4 n)L C S 2L,

where the desuspension acts in the cohomological degree t. Only certain of the external

homotopy operations 7rV -+ 7r"S2 V defined in 5.2 factor through 7rE-lq2F (n)V, and

similarly for the operations of 5.3. One readily checks that the operations that factor in

this way are:

t : 7TrnV - rn+i( l2FZ()V)

defined only when 0 < i < n and i, Si, ... , Sn are not all zero, and

V : q2 F'(n+1)(7r*V) -- + 7r(q2 F"(n)V).

The resulting operations on rL, for L C sZ(n), are right A-operations

(-)Ai : ((rsn+1L),..,24 (IrSn+1+i( 7 lq 2 FC n)L))fs,..,2 1 7.([] + 1 + L)2 ,.,2si

defined whenever 0 < i < sn+1 and not all of i, Sn,..., si equal zero, and a bracket:

:((7rL)+,. 0 (7r*L)t+ - + (q 2 F4(n+1) 7rL)t+t' +1, ...,81+'

S (,7r~q2F' () L)t+t D (7r*L)t+t'+1 Ssn+i + sn 4,...S1+ s sn+ 1+ 8 , , +-~-I n+1 .. )81(~)~,

We have written the bracket as a map from (7rL)®2 to clarify the degree shift, but nev-

ertheless, the top A-operation, whenever it is defined, acts as a restriction for this bracket.

Indeed, this set of natural operations satisfies the evident modification of Proposition 5.7

(c.f. Proposition 7.1).

100

7.3. The category UI(n +1) of unstable partial right A-modules

For n > 0, let U(n + 1) denote the category of unstable partial right A-modules, the algebraic

category whose objects are vector spaces V E V+ 1 equipped with linear right A-operations

i -) : 1 V , I ,..., -+1 V,+1+i,2Sn,...,2s1

defined whenever 0 < i < sn+1 and not all of i, sn,.. , si are zero, satisfying the unstable

A-Adem relations of Proposition 5.7(5).

We have shown that an object of 7rL(n) is in particular an object of U(n + 1), indeed,

the U(n + 1)-structure on 7rL consists solely of its non-top A-operations, which are linear

as required.

7.4. The category W(n + 1) of Z (n)-f-algebras

For n > 0, let W(n + 1) denote the algebraic category whose objects are V++ 1-graded

vector spaces which are simultaneously an object of U(n + 1) and of (n + 1), such that

the compatibilities of Proposition 5.7 are satisfied. Explained another way, an object of

W(n + 1) is such a vector space with the bracket and all of the A-operations (both top and

non-top) described in 7.2, subject to the compatibilities of Proposition 5.7.

This category has a number of useful properties, following from the calculations of [13],

primarily:

Proposition 7.1. The operations defined in 7.2 generate the set of natural operations on

the homotopy of simplicial objects of L(n) and satisfy the compatibilities of Proposition 5.7.

The category W(n + 1) is isomorphic to the category irC(n) of C (n)-I-algebras. The monad

Fw(n+l) on V++ 1 factors as a composite Fu(n+1) o FL(n+l), with monad structure arising

from a distributive law [2! of monads on V+ 1.

Proof. All of these facts are easy to prove after observing that, for W E sV a coproduct of

spheres, ir,(FC(n)W) embeds in 7r,(F(Y)W), which, along with 7r.(A(Y)W), is described

in Proposition 5.9 (although by A(Y) and F(Y) we mean the shifted monads of 7.1). In

order to make this observation, let write Wo for t. W ..,O, the non-restrictable part of

W. This is actually a sub-coproduct of W, the coproduct of those summands of W which

lie in homological dimension (0,..., 0). There is a commuting diagram of simplicial vector

spaces, containing two short exact sequences:

fi (Y0(A Wo >A0

0 > A(Y) WO > (Y)WO

101

where a and 6 are inclusions and -y and p are epimorphisms. On homotopy groups: 0, is

injective (its source and target are well understood), so that p. is surjective. Thus y, is

surjective (after all, -y is an isomorphism in those internal degrees in which its codomain in

non-zero), implying that (p-y). is surjective. This implies that a. is injective, as hoped. E

The following two lemmas are the direct analogues of Lemmas 6.10 and 6.11:

Lemma 7.2. For n > 0, the monads Fu(n+1) and Fw(n+1) on y+ may be promoted to

monads on the category qVk+1 by requiring that quadratic gradings add when taking brackets

and double when applying A-operations. The same holds for C(n) for n > 0.

7.5. The factorization QZ(n) 0 QU(n) of Qw(n)

For n > 0, we define

QU(n+1) := (W(n +1)f U(n + ) __4 +-

That is, for X E W(n+1) we may take the quotient by the image of the non-top A-operations

(which are linear, so that this operation is well defined). In fact, it is not hard to see that,

for X E W(n + 1), QU(n+l)X retains the structure of an object of Z(n + 1), so that we may

view Qu(f+1) as a functor W(n + 1) -* Z (n + 1).

Lemma 7.3. For n> 0 and X C W(n + 1), X is in particular an object of Z (n + 1), and

the vector space Qu(n+)X retains this structure, yielding a factorization:

QW(n+l1) - QC(n+l) 0 QU(n+l) : (W(n + 1) -- 12(n + 1) -+ 7 ++

Moreover the composite QU(n+1) o Fw(n+1) equals the free construction FL(n+1).

Proof. Similar to the proof of Lemma 6.11, using the observation from the proof of Propo-

sition 7.1 that 7r,(F'C()W) C ir*(Fr(Y)W). El

This differs from the definition of QU(o) -+ Z(0), in which one takes the quotient by

all the P-operations. Indeed, the category W(0) differs from the categories W(n + 1) (for

n > 0) in a number of ways, primarily because W(0) is a category of cohomology algebras

while the W(n + 1) are categories of II-algebras. If X C W(0) and Y E W(n + 1):

(1) Y is a Lie algebra, while X is only an S(Y)-algebra;

(2) the P-operations on X are always defined and vanish when out of range, while the A

operations are simply undefined when out of range;

(3) the top P-operation is the self-bracket and thus is linear, while the top A-operation is

the restriction and thus a quadratic refinement of the bracket.

102

Nevertheless, the two regimes share the following common ground:

Corollary 7.4. For all n > 0, there are algebraic categories W(n) and U(n), a forgetful

functor U' : W(n) -+ U(n), and a functor Qu(n) : W(n) -+ L(n), such that

UZn) o Qu(n) _ QU(n) 0 U', QW(n) - Ql(n) 0 QU(n) and QU(f) o FW(n) = F ").

7.6. Decomposition maps for C(n) and W(n)

Here we will introduce decomposition maps for the categories ((n) and W(n), and calculate

the resulting quadratic part maps. The definitions are simple enough, and the reader can

verify that each is well defined. For any n > 0, the following formulae define decomposition

maps je : QC(X Y Y) - QCX ®QCY.

jw(o) : P9 Z2Pi[zi,- ,Zaj H Zi®Z2, if f = 0, a = 2, z1 E X, z 2 E Y,

0, otherwise.

I .w(n+1) :[zi,--, za]Ai, -A, F+ Z1 0 Z2, if e=o 0, a = 2, z, E X, z2 E Y,

0, otherwise.

[-z] Z1 () Z2, if r = 0, a = 2, zj E X, z2 E Y

0, otherwise.

Proposition 7.5. Suppose V E V+. Then:

(1) quc(n) is the composite F")V - q2F(n)V C ES2 V;

(2) quw(,) is the composite Fw(n)V -- FG(n)V -+ q2F"()V C ES 2 V.

Proof. Consider the case W(n + 1) for n > 0. As que vanishes except on quadratic grading

2, one only checks terms [x, y], X[ 2 ], and xXi (a U(n + 1)-operation, not the restriction):

quw(n+l)([x, y]) = jW(n+l1 )([X + X2, Y1 + Y2] + [Xi, Y1] + [X2, Y21)

jW(n+l)([X1, y2] + [X2, y11) =x X 0y + y 0 X,

which is precisely the representation of [x, y] in q2 FZ(n+1)V C E(V0 2 ). For similar reasons,

quw(n+l)(XA/) vanishes (as U(n+1)-operations are linear), while quw(n+l 1 )(X[2) equals x &x

as desired. The other cases, including the case of W(0), are barely any different. D

103

104

Chapter 8

Operations on W(n)- and

U(n)-cohomology

8.1. Vertical 3-operations on H, and Hj(O)

We will now define natural homomorphisms for V E V+

0i : (FW(O)V)t+i+1 - Vt, for 2 1, the degree m, quadratic grading 2 part q2 (FW(o)V)m decomposes as

q2(FW(O)V)m = im((S2 V)m- 3' q2FW(O)V) e E im(Vm-i- q2 Fw()V).

2<i<(m-1)/2

Moreover, each map Pi : Vt -+ q2 (FW(O)V)t+i+l appearing in this decomposition is an

isomorphism onto its image, so that for 2 < i < t we may construct 64 as the composite

Vt ).

Here we have projected onto the quadratic filtration 2 part, and then further onto the relevant

summand in its natural decomposition. Note that although Pt : Vt - q2 (FW(O)V) 2t+l is

a non-trivial linear map when t > 2, its image is entangled with the image of the bracket,

and we are not able to split it off. Thus we are not able to improve on the bounds 2 D

(FW(o)DV)t+i'+

Proof. When V is of finite type, as Fw(o) preserves vector spaces of finite type, we may

simply define Ot to be the dual of Oi. This is natural on vector spaces of finite type, and any

vector space is the filtered colimit of such.

In fact, whenever t > 2 we may define a non-linear function

*V -+ (CHiom-coalgV)

which completes the collection of functions 0*, but not by this method: we use the upcoming

Proposition 11.14 to define this top Ot. That we need to do this is a disadvantage of working

with cohomology algebras, as opposed to homology coalgebras.

Proposition 8.2. Suppose that X C sW(O) is almost free, so that we may identify H'Q(0)X

with 7r*DQW(O)X. Then for 2 < i < t, the chain map 0- of Proposition 6.4 induces a linear

operation

JY : ( H, O X| -+ ( H 0 X )s++1-

These operations are natural in maps preserving the generating subspaces, and satisfy the

unstable 6-Adem relation of Proposition 5.3(6).

If X is of finite type, this statement may be amended to include a (potentially non-linear)

operation

6v : ( H,(O X )O -+ ( H, X )'+

induced by the function 0*.

For any X E sW(O), the bar construction Bw(o)X of X has a natural almost free structure,

so that Proposition 8.2 may be used to construct natural operations on HN(0 )X.

Proof of Proposition 8.2. The finite type assumption is needed in order to define the oper-

ation 6, as it is not induced by a chain map on NQw(O)X. Instead, it is induced by the

potentially non-linear function

*(N 0 pHWom-coalg DX)t -- (N1 prHom-coalg DX) 2t+1

106

induced by the function 0* defined using Proposition 11.14. We will give the proof that

the operations 6i with 2 < i < t are well defined and satisfy the 6-Adem relations, in the

cohomological variance. By working with homology coalgebras, and thus avoiding double-

dualization, the proof given below extends to encompass the extra operation 6R in dimension

zero. This exercise is left to the reader.

The conditions of Proposition 6.4 are satisfied with 6 = Oi and G is the identity functor.

The condition

0 0 AV = 0 o EFCV + 0 o Focv : FO Fe V -- + V

just states that given an iterated expression in FCFCV, the two obvious ways to produce a

summand of the form P'v under the map [ : FOFeV -+ F2V are the only ways, due to

the homogeneity of the P-Adem relations.

It just remains to prove the 6-Adem relations, which we will do using the technique of

[47], the point being that the algebra of 6-operations is Koszul dual to 'P. For this, we define

a map Oij, whenever i < 2j, 2 < j < t and 2 < i < t + j + 1:

Oij (Fw(o)yV)t+i+j+2 -R (q4Fw(O)yVgt+i+j+2 im(Pidi) (P V ,

where we have split off the image of PJi = P-Pi as before. This is possible since neither

Pi nor P' are entangled with the bracket in these ranges. We may identify Qw(o)Xs with

V, at the cost of replacing do with c o do, as in Lemma 2.6. Define 6O, to be the composite

V,+1 - FV -! V. This will be the nullhomotopy giving the 6-Adem relation. As in the

proof of Proposition 6.4, we have dk 0 a =Vi o dk+1 for k > 1, and Oij has nullhomotopy

the sum

edoOij + 6 3 (Edo + di) (cdo6ij + 6ij doE + 6ij do)do.

The 6-Adem relation will follow from

OiTdo = (cdo~ij +6ido6 + Z ( do6 :) FVs+1 -+ Vs,

This identity states the following: if V e V+, and f(g) is a nested W(O)-expression with

gk C Fw(o)V and f(9) c Fw(o)Fw(o)V, then if we write do : Fw(o)Fw(o)V -+ Fw(o)V

for the monad product map, there are only three ways that one may obtain expressions of

the form P2 Piv in do(f(gk)): for some k, A = P'Piv, and f adds no further operations

to this term; f = PiPjgk for some k for which g = v is a unit expression; or for some k,

9k = POv, and f has Pa(g) as a summand. In this last case, after applying do, we may

need to rearrange the composite P'P'v using the P-Adem relations, and we sum over those

(a, 3) producing a summand P'Piv.

107

This shows that the proposed nullhomotopy equals

and as Goerss [33] constructs the P-algebra as the Koszul dual, in the sense of [47], of the

A-algebra, so that (a, /) -+ (i, j) if and only if (i, j) A (a, 3), so that the nullhomotopy

equals the desired sum:

Z~,)Ac)0,3 0 a El

The same constructions work in the category sU(O) of simplicial unstable '-modules, the

only difference being that when we define Oi, we need not worry about Lie algebra structures,

and we can define a map

Oi : (FU(o)V)t+i+1 -- + Vt

whenever 2 0.

It will be useful to encode this structure in a definition. Write Mv(1) for the algebraic

category whose objects are vector spaces M E V1 with left 6-operations

JV : Mt -+ M+ 1+ defined whenever 2 < i < t,

satisfying unstable 6-Adem relations analogous to those of Proposition 5.3(6).

Proposition 8.3. Suppose that X E sU(O) is almost free. Then the chain maps Oi of

Proposition 6.4 give Hj(O)X the structure of an object of M,(1), natural in maps preserving

the generating subspaces. In fact, M,(1) is the category of U(0)-H*-algebras.

See 9 for further discussion of this fact, and Proposition 9.1 for a restatement. It is not

true that H, () is an object of M,(1), a fact that we emphasize because H n will be an

object of My(n + 1) for n > 1 (under definitions made in 8.2).

In order to give a basis for a free object in M,(1), for a sequence I = (ii,... ,i) of

integers ij > 2, we use the function

TE(I) := max{(ii), (i 2 - ui - 1), (i 3 - i2 - i - 2), ... , (i -- -+ 1)},

of 6.6, following the convention that max(0) = -oo, and the notion of 6-admissibility from

5.4: each ij 2 and ij+1 > 2ij for 1 < j < i.

Lemma 8.4. For V C V1 with homogeneous basis B, a basis of FMV (l)V is

{6'b b E B', I 6-admissible with Ti(I) < t}.

108

We will often apply such results as these when V is concentrated in degrees Vo. At this

point we introduce a notational abuse, identifying V+ with the full subcategory of V' with

objects concentrated in these degrees. The effect of this will be that we will be able to write

FM-(l)V for V E V+. Restricting Lemma 8.4 to such cases:

Corollary 8.5. For V c V+ with homogeneous basis B, a basis of FM-(l)V C V1 is

{ J'b Ib E Bt, I 6-admissible with TiT(I) < t}.

8.2. Vertical Steenrod operations for H, and H( when

n >1

For V E V+, we will define natural homomorphisms

0': (Fw(n)V)2t+ 1 . tSn+i-1,2sn-1,...,2s1 Sn,---,S1

which are defined for all i, si, ... , Sn > 0 and t > 1, but are zero except when 1 < i < sn

and not all of i - 1, Sn-i, ... , si are zero. These are rather easier to define than in the n = 0

case investigated in 8.1, as the monad FW(n) is a simple composite Fu(n)F4(n) of monads

when n > 1. Indeed, there are natural monomorphisms

(-)Ai_. : Vt (q2 FW(n)V)2 +l-12s_..2s1

defined only when 1 < i < n and i - 1, Sn-1,..., si are not all zero, and an inclusion

incl : q2F()V --+ q2 FW(n)V.

As in the n = 0 case, the images of the listed maps are linearly independent and span

the quadratic grading 2 part of Fw(n)V. We define 6 to be zero unless 1 < i < n and

- 1, sn1, .. , sI are not all zero, in which case we define it as the composite:

0i : ((FW(n)yV)2t+1 pr im( A_1) -' VtSn+i-1,2 --1,..,2s1 ! o' im(A ) ( n,... ,

One can give exactly the same definitions for the free construction in 11(n), producing

functions 6 : FU(n)V - V which are zero under the same conditions as for W(n).

Write MN(n + 1) for the algebraic category whose objects are vector spaces M E V'+

with left Steenrod operations

Sqi : MSn+1' ... S -- MS+1+1,sn+i-1,2s,_1,...,2s1M; 2t+1

109

which are zero except when 1 1) there is no disparity between the unstableness conditions on

W(n)- and U(n)-cohomology, so that the analogue of Propositions 8.2 and 8.3 is:

Proposition 8.6. Suppose that X E sC is almost free, where C stands for either W(n) or

U(n) with n > 1. Then the -chain maps O' of Proposition 6.4 give H*X the structure of

an object of M,(n + 1), natural in maps of almost free objects preserving the generating

subspaces. Again, Mv(n + 1) is the category of U(n)-H*-algebras.

In order to give a basis for a free object in Mv(n + 1), for a sequence I =. . . ,1i) of

integers ij > 1, we define

mr(I) := max{(ii), (i 2 - i1 + 1), (i 3 - i 2 - ii + 2), ... ,- - - - ii + ( - .

Recall that I is Sq-admissible if each ij 1 and ij+1 > 2ij for 1 < j <

Lemma 8.7. For V E V'+1 with homogeneous basis B, a basis of FMv(n+l)V is

fSqi b C BP""'8'', J Sq-admissible with m(J) Sn,if sn-1= - =s=0 then J does not contain 1

Performing the same abuse of notation as in Corollary 8.5:

Corollary 8.8. For V z Vn with homogeneous basis B, a basis of FMv(n+l)V Vn+1 is

b B B" '..S'1, J Sq-admissible with rm(J) Sn,Sqib

V if sn-1=- -s1=0 then J does not contain I

8.3. Horizontal Steenrod operations and a product for H(

For any n > 0, we will construct operations on the homology H,(n) arising from the S(M)-

algebra structure or Lie algebra structures.

Indeed, suppose that X E sW(n) is almost free. Then Qu(n)X E sfC(n) is also almost

free, on essentially the same generating subspaces. Thus, the cohomotopy of Qw(n)X =

QL(n)QU(n)X is an instance of simplicial partially restricted Lie algebra cohomology. Coho-

mology operations of this type are discussed in 6.8 and Appendix A. In the present context,

we have two equivalent definitions, one using On(n) and one using OW(n), and until Appendix

110

A, we will use VW(n), defining operations

Sq: H~n)(X) 2 7rn+j-DS 2QW(n)X H (X) and

p : H X H (X) L-4 7rnl+n2DS 2QW(n)X * Hgj(l2+1(X)).

In more detail:

Proposition 8.9. Fix n > 0. For X E sW(n), there are natural operations

Sqh : (H (n)X)sn+1.Si -- + (H- ) + 2sI

...~...Sl ® (H{l(P)X)+1+sn+Pn" ... .S1+P1y:(H Xn),+''." t (H' nX)Pqn+1',.. P1 --+4 (H' (n X)""q+1+,9,.,19

with the following properties

(1) the product ft gives H*L the structure of a (grading shifted) S(W)-algebra;

(2) the squaring operation on H*L equals the top Steenrod operation:

q 1 +1+ 2 for x E (.S) 1

(3) if x c ( H lX)sn+1,.iS, then Sq'x = 0 unless 1 2;

(5) every Steenrod operation is linear;

(6) the Cartan formula holds: for all x, y c H*L and i > 0,

Sq'(xy) = -_l 1(Sqkx)(Sqiky);

(7) the homogeneous Sq-Adem relations hold, making H*L a left A-module.

Proof. Almost everything here follows from Proposition A.3, which demonstrates that the

operations we are discussing here coincide with those defined on H , (c.f. Propositions 6.12

and 6.13). The same technique used to prove Proposition 6.14 proves the new part of (3).

For (4), when n = 0, (3) shows that Sqkx = 0. On the other hand, to see why Sq2x = 0

when t > 2 is more difficult, especially since we have not determined the category of W(0)-

UI-algebras. Nonetheless, as in the proof of Proposition 6.14, we will prove this for the

universal example 4z E H,(0)KW('('O. The reverse Adams spectral sequence ( 3.4) can be

equipped with a quadratic grading if we view the generator of K. as lying in quadratic

grading one, and is of the form

gkE = qk(H,(K" "W'')P = qk(H )K(O) )T.

111

As Sq2 t E q2 (HOj2)K, 8't) 2 t+l, we need to determine

q2H2 k7rW(O),) t+1 and q2 1 K7rw()), ' ,+1q W(HwO)KO,s )2t+ and ( ()KO,, )2t+1,

and as in the proof of Proposition 6.14, we need to see how far F2 {z } is from being free in

7rW(0). Fortunately, we only need to answer this question in quadratic grading two, and

Ferw(O) (F2{f Z}) =r, (SW(O),t) = ir*(FW(O)Kt).

Now if t > 2, q2 FW(O)V naturally decomposes as

q2 FW(O)V = q2 FS()V ED p 2V .. .D pt-1,

and we calculate, by Proposition 5.6:

q2 Fw(O)(F2{Its})+ =F 2 {A 2 1tAzt,... , Ast}

That is, there are two missing A-operations, AO and A 1 , in the functor Fs(2), and the

presence of the operations P2 . pt" do not effect q2FarW(O) (F2 {zt}) in internal dimension

2t + 1. We now have enough information to proceed as in the proof of Proposition 6.14,

since Akit E q2F'w( )(F2{Zt }) + for 2 < k < s. F-1

For n > 0, write Mh(n + 1) for the algebraic category whose objects are vector spaces

M C V++1 with left Steenrod operations and a commutative pairing satisfying the conditions

of Proposition 8.9. We have simply shown that H takes values in Mh(n + 1) - it is

certainly not true that M 1,(n + 1) is the category of W(n)-H*-algebras, as we have also seen

that H n takes values in Mv(n + 1).

Note that the unstableness condition implies that x = 0 whenever x E M 0 0 . Indeed

Proposition 8.10. Suppose that n > 1, and that V c V"'1 has homogeneous basis B. Then

FMh(n+l)V is the quotient of the non-unital commutative algebra

b E B "l, 1 '-', l J Sq-admissible with e(J) s +1S(%) Sqb

if s = - -- = si = 0 then J does not contain 1

by the relation b2 = 0 if b e B ,0... Here, e (J) :=j - je- - - ji is the Serre excess of J.

Proof. By [48, 6.1], the true free object is a quotient of what we propose. It is in fact equal

to what we propose, because the two-sided ideal in the homogeneous Steenrod algebra A

generated by Sqh is spanned by those admissible sequences ending in Sq, so that forcing

Sqh = 0 in the relevant degrees has no unintended consequences. E]

112

Corollary 8.11. Suppose that n > 1. For V E Vn with homogeneous basis B. Then

FMh(n+l)V E Vn+1 is the non-unital commutative algebra coproduct

b E BS""'.'S(e) b t u A(W) [b b E Bt0]

Sn, .. ,S1 not all zero

8.4. Relations between the horizontal and vertical operations

It will be helpful to be able to reduce expressions in the various available operations to a

standard format,

Hk SqJv {3)xk when n = 0, or Il SqjkSql xk when n > 1,

which is possible, thanks to:

Proposition 8.12. Suppose that x, y E H (o)X. If Sqhx E (H, (o)X)s, then JvSqhx 0

for 2 1 and X, y E H (n)X. Then SqvSqhx = 0 and Sq"(xy) = 0.

Proof. For the case n = 0, suppose that X E sW(0) is almost free on generating subspaces

V,. It is enough to prove that the composite

N,9+1(Qw(O)X)t+'+1 --O N_(M(+X Ns_1(S2(QW(O)X))t-1

is nullhomotopic, using a similar method to that used in the proof of Proposition 8.2. For

any V E V+, there is a natural composite

(S2V) - (q2Fw(O)V)t-Pi(q4Fw(O)ygt+'+1,0e

whose maps we have labeled a and 3 for convenience. The map #3 im(a) is not a monomor-

phism when i = t - 1 is even, as in this case, for any v E Vi/2

Pi[vv] = PiPi/2v = zi2-2, (2 (k-i/2 -1) pi/2-kpkV - 0,

as each expression PkV in the sum vanishes by the unstableness condition. However,

ker (O im(o)) is contained in ker (quw(o)), and im(3 o a) does naturally split off as a direct

summand of (q4 FW(O)V)t+i+l. We write hi for the composite:

hi:((Fw()ye+i+1 pro (im(Q 0 k))er1 im(a) u (2)-r(/) rn im(a) (S2V)

Identifying QW(O)X, with V, as in the proof of Proposition 8.2, the nullhomotopy associated

113

do -with the composite iSi : (V+, -:4 FV -4 V) is the sum

(cdohi + hidoe + hido)do,

and the relation we seek will follow from the identity

hido = (cdohi + hidoe + quw(o)doOi) FVs+1 -+ Vs ,

as then #wto)Oi = quw(o)doOido = dohi + hido. This identity states the following: if V E

\?+, and f(gk) E Fw(o)FW(o)V is a nested W(0)-expression in various expressions gk E

FW(o)V, then if we write do : FW(o)FW(o)V -- + Fw(o)V for the monad product map, there

are only three ways that one may obtain summands of the form P'[vi, v2] in do(f(gk)) E

(FW(o)V)t+i'+: for some k, gk = Pi[vi, v21, and f adds no further operations to this term;

f = Pi(gki, gk2], where 9k, = vi and 9k2 = V2 are unit expressions; or for some k, gk = [vi, V2],

and f has Pi(gk) as a summand.

For the case n > 0, the proof only becomes easier, the main difference being that in the

corresponding composite:

(q2FLn)3)t-+-(q2FW(O)y)t 1(q4FW(O)y)t+'-1,a '

both ar and 01im(a) are monomorphisms.

8.5. The categories Mhv(r + 1)

For n > 1, let Mhv(n-+ 1) be the following algebraic category, monadic over V'+1 . An object

of Mhv(n + 1) is a vector space V E V+ 1 which is simultaneously an object of MV(n + 1)

and of Mh(n + 1), and in which

Sq ,(xy) = 0 and Sq' (Sqh(x)) =0 for all xy e V.

By Proposition 8.12, for n > 1 and X E W(n), HIQWX is naturally an object of Mhv(n+ 1).

We will prove in Corollary 14.7 that for n > 1, Mhv(n + 1) is the category HWom of W(n)-

cohomology algebras.

The corresponding facts are not true for n = 0, so we do not even define a category

Mhv(1)-

For any n > 1, the monad FMhv(n+l) factors as FMh(n+l)FMv(n+l), with the evident

distributive law of monads, and combining Corollary 8.8 and Proposition 8.10:

Corollary 8.13. For n > 1 and V E Vn with homogeneous basis B, FMhv(n+1)V is the

114

quotient of the non-unital commutative algebra

b E Bn'.Si, I, J Sq-admissible with m(I) 5 so, e(J) I~

S('W) SqjSqlb if sn-- s - 1=0 then I does not contain 1if sn- - ==O then J does not contain 1

by the relation b 2 = 0 if b E B'.

Although we do not define JVMhv(1), it will be useful to have a description of the composite

FMh(l)FMv(l). Combining Corollary 8.5 and Proposition 8.10:

Corollary 8.14. For V E V+ with homogeneous basis B, FMh(l)FMv(l)V is isomorphic to

the non-unital commutative algebra coproduct

S (le) FSJvb b E Bt, I non-empty, 6-admissible with TE(I) <,1 A(e) [b b B].

I h I J Sq-admissible with e(J) LI, and 1 V J JFor elements b 1,... ,bN of B with bk E Btk and appropriate sequences Ik, Jk, we have

HiN Sqjk6v bk E (FMh) FMv(1) V) +-k(nJ+fIk+1)k=1 h Ik -1+Ek(2'k(t-nI-k+-k+1))

8.6. Compressing sequences of Steenrod operations

The following theorem creates a model for the convergence of a spectral sequence which we

will discuss in 13. One should think of FMhv(n+1)V as the Eoo-page of a first quadrant

cohomotopy spectral sequence and FMh(n)V as the cohomotopy of the total complex.

Theorem 8.15. Suppose that n > 1 and V E V' . Then there is a decreasing filtration on

FMh(n)V, the target filtration, and an isomorphism

f : (FMhv(n+1) V)Sn+1- - S1 [EoFMh(n) V]n+1.-S

defined by requiring that f (Sq~v) = Sqlv for v G V, that f (wIw2 ) = f (wi)f (w1) for WI, w2 E

FMhV(n+l)V, and that

f(Sq~w) = SqI+Snf(w) for w e (FMhV(n+l)V)n+l.-S

Proof. The proposed map f is not a well defined map to FMh(n)V since the Adem relations

between the Sqh are not preserved by the proposed map f. Write W(V) for the quotient of

S(') [A 9 FMv(n+1)V] by the horizontal unstableness relations and Cartan formula, so that

FMhv(n+l)V is obtained from W(V) by taking the quotient by the two-sided ideal generated

115

by the horizontal Adem relations. Then may define a map f : W(V) -+ FMh(n)V by

requiring the same of f as of f. There is a decreasing filtration on W(V), given by

FPW(V) = e $W(V)"+1'.-'

sn+1iPSn,---,s1 o t>1

and we define the target filtration on the target by FP(FMh(n)V) := f(FPW(V)).

The map 7 fails to descend to a well-defined map FMhv(n+l)V -- FMh(n)V, because it

does not annihilate the Adem relations. However, we will show that it does send them into

higher filtration, so that 7 induces a well defined map f as advertised: if w E W(V)s"'--.'I1

and i < 2j, then

7(SqSqhw - Z/j (-k )Sqi+j-kqk

which is in filtration Sn+1 + + j+ 2(s + 1 - k) > sn+1 + i + j (the second equation holds

by simply shifting the dummy variable k, the third by an Adem relation in the codomain).

What remains is to show that f is an isomorphism as in the theorem statement, which

we approach simply by choosing a set of multiplicative generators for both the domain and

codomain. The domain is generated by those expressions SqiSql'v, for v E V,"'''" running

through a basis of V, and appropriate Sq-admissible sequences J and I. The codomain

is generated by expressions Sqjfv, for v E V~ia"' running through a basis of V, and

appropriate Sq-admissible sequences K. It is a combinatorial exercise in the properties of

admissible sequences to show that these sets of generators are put in bijection by f, and

this bijection sends polynomial generators to polynomial generators and exterior generators

to exterior generators. L

116

Chapter 9

Koszul complexes calculating

U(n)-homology

We will now discuss the Koszul resolutions that one may use to calculate HY*(4X for X a

(non-simplicial) object of tL(n) or W(n) of finite type, using Priddy's technique [47], adapted

to an unstable context, as in [20] and [33, Chapter V].

9.1. The Koszul complex and co-Koszul complex

Write N* and C* for the chain complexes N QU(n)BU(n)X and C*Qu(n)BU(n)X. We will

use the convenient bar notation after which the bar construction is named, c.f. [28, 7].

Suppose that n = 0, then the vector space N, is spanned by

[Pik,+ .+ki ... Pik,-1+---+ki+l . .. I ik2+ki .. . Pik1+1 Pikl i l . ..1

where x E X, the expressions in each of the r spaces are P-admissible, and none of the

spaces is empty (so that kj > 0 for 1 < j s). Such an expression represents an element

of the repeated free construction QU(n)B, -(n)X (Fu(o))sX, with the requirement that no

space be empty reflecting having taken the quotient by degenerate simplices. In particular,

this expression equals zero unless 7ff(ik,.-ki, - ii- ) < lXi-When n > 1, the vector space N.+ is spanned by expressions

again without empty spaces, and subject to an admissibility condition. However, these

expressions are only defined when every A-operation appearing is defined. That is, if x c

X , then we require m(ik+.+i.ki, -. -i) 5 sn, and for no AO to appear if s_ - -=

117

si = 0.

Each of these complexes admits an increasing filtration, the length filtration, with FiN,

generated by those terms in which ik+ .+ki K < for each term, which is to say that there

are at most f generators appearing in the s free constructions in C, = QU(n)B, ()X e

(Fu(n))sX. Note that Fs_1 Ns = 0.

Write E'8 for the spectral sequence of the filtered complex NIX, so that EO is the

associated graded complex. As F,-1N,+ = 0, E' = 0 for f < s, and E0, is the subspace

F,Ns+ of N+. Priddy [47, Proof of Theorem 5.3] shows that E = 0 for f > s. Thus, the

groups

KUn)X := El , equipped with d : E1 ,s - E1_1,s_1

form a subcomplex of NL, the Koszul complex, whose inclusion is a homotopy equivalence,

and El8 is the preimage of F,_ 1 N, 1 under

d :F, N,+- - Fs N8+_1

Rather than determining these groups directly, Priddy works with their duals, K (,) X,

which form a cochain complex with homology Hj (n)X. In fact, Priddy's theory shows that

the cochain complex K(n) X, the co-Koszul complex, is actually a differential unstable left

module over the same operations as its cohomology H (n)X, and indeed that this (partial)

module is free. More precisely, Ko ")X = X, and K () X is free on the subspace X* of

K XA U(n)

Proposition 9.1. Suppose that n > 0, and X is an object of U(n) of finite type. The chain

maps Oi (OI when n = 0) on C*Qu(n)BU(n)X restrict to the subcomplex K X, and induce

an Mv(n + 1)-structure on K (fl)X which commutes with the differentials. The inclusion

DX 2 K X C K* X induces an isomorphism FMv(n+l)(DX) - K 2 X. Moreover,U(n) - unthis Mv(n+ 1)-structure on K (n) induces the Mv(n+ 1)-structure on H (n X of Propositions

8.3 and 8.6.

Although it is easier to calculate the co-Koszul complex, we will need to understand the

Koszul complex itself in order to calculate the W(n + 1)-structure of H . For this, we

will introduce a little notation:

Proposition 9.2. Suppose that X c U(0) has homogeneous basis B. Then K* 0 )X has

basis

{,v*b I bE B', I 6-admissible with TiT(I) t},

118

where we define, for any x G Xt and I 6-admissible with Th(I) < t:

6'*x := - ~ [Pke Pki] x for x E Xt.

If X is of finite type, there is a basis {3J(b*)} of K (O)X constructed using the isomorphism

of Proposition 9.1, Corollary 8.5 and the basis {b*} of DX dual to B. The bases {6'(b*)}

and {6f*b} are dual.

The differential of K 1 0 X is given by the formula:

)= ( 3 (k ,...,k 2)

(kg,...,k 2 ) A-admis.

Asumming over those K = (kj,... , k 1 ) such that (kt,..., k 2 ) is 6-admissible, and yet K -4 I.

Note that the sum defining 6'*x is finite, simply because the A-algebra is graded by the

sum of indices. We may assign 6v*x = 0 for x E Xt and 7(I) > t, if we wish, since:

Lemma 9.3. If I + J, then m(I) m(J). This inequality is strict if I and J have length

2.

In fact, this lemma ensures that the sum defining 3J*x is finite. That is, any K with K I

must have m(K) > m(I) 6-Adem relations only decrease Wi. Indeed, we may further restrict

the two sums appearing in this proposition be requiring that 7T(K) t in each case, but

this has no effect. Dually, in the co-Koszul complex, the operations 6Y are undefined when

out of range.

Proof of Proposition 9.2. Firstly, we may assume that X is of finite type, as any object of

1(O) is the union of its subobjects of finite type, and the functor K() preserves unions.

It is enough to check that 6J*b is in fact a member of N;, not just of C", as then the

collection 6J*b will evidently be the dual basis to the 67(b*): in the sum defining J*b, the

only 6-admissible sequence K appearing is K = I.

Using [47, Lemma 3.2], to check that 6v*b E N*, we only need to check that d(6v*b) E

F,- 1 ,_1. To check this membership condition is to check that 6v*b pairs to zero with

im(d* : D(FsN8 1 ) -- + D(FNS)). Priddy's proof shows that D(FN) is spanned by

functionals [(Pks)*1 ... (Pkl)*]b*, which pair with the 6)*c according to:

[ ) (Pk)*]b*) (3J*c) = b*(c) - (61 coeff. of 6K E A written in admissibles) .

However, the image of d*, as determined by Priddy, is spanned by the space of '3-Adem

relations' (see [47, Theorem 2.5 and proof]), and these tautologically evaluate to zero on any

119

Jv*b.

The same analysis applies in the n > 1 case. Although we write the bar construction

on the right, we end up with a left action of the homogeneous Steenrod algebra, as the

homogeneous Steenrod algebra is Koszul dual to the opposite of the A-algebra with an index

shift.

Proposition 9.4. Suppose that n > 1 and X C U(n) has homogeneous basis B. Then

K* *X has basis

S*b b ,B , J Sq-admissible with n(J) < sn,if sn-1- - =- =0 then J does not contain I

where we only define

Sqj*b := b [4s 11- - -JAk,_1]K - J

when J and b satisfy the conditions on m(J) and on the appearance of 1 in J. If X is of

finite type, this basis is dual to the {Sqib*} basis of K (n)X constructed using Proposition

9.1, Corollary 8.8 and the basis {b*} of DX dual to B. The differential of K* is given

by the formula:

d(Sql*x) = Sqk..k2)*(xAkl)KIJ

(kR,...,k 2 ) Sq-admis.

summing over K = (ke,..., k1) such that (kj,..., k 2) is Sq-admissible and yet K 4 J.

As part of the omitted analysis, we would use 5.8, and the fact that the A-algebra and the

homogeneous Steenrod algebra are Koszul dual, to show:

Lemma 9.5. If I and J are sequences of non-negative integers (of any length), such that

J 4 I, then m(J) m(I), and if 1 appears in J, it must also appear in I.

This implies that all the summands in the above definition of SqJ*x are indeed defined.

9.2. The W(n + 1)-structure on H i(n)X

Suppose that X C W(n) for some n > 0. The form of the bases of Ku(*)X given in

Propositions 9.2 and 9.4 imply:

Corollary 9.6. The Koszul complex K *X is naturally a subcomplex of N-QU(n)BU(n)X.

There are thus two monomorphic quasi-isomorphisms of chain complexes with homology is

H* and we denote their composite :

K: (Kn(")X C N;-QU(n)BU(n)X C N-QU(n)BW(n)X).

120

El

The key upshot of Corollary 9.6 is that cycles in the Koszul complex map to normalized

cycles under 3.

Now H,,)X is an object of W(n + 1), since it can be calculated as the homotopy of

QU(n)Bw(n)X E sC(n), and this structure will be needed for the composite functor spectral

sequences discussed in 12.1. We will go some way to calculating this structure in this

section. Our method will be to take cycles in the Koszul complex, map them into the large

complex using 3, perform the operations in question, and then homotope the outcome back

into the Koszul complex.

We will need a little notation for elements of the various bar constructions. We will label

the s + 1 free constructions in B, with subscripts in angle brackets:

BW(n)X = F W(n) F F)...F )XS (-1) (0) (8-1)

so that we can then indicate in which free construction operations are being performed. For

example, when n = 0 and x, y E X, Bj(0 )X contains an element

P' 4~ Pf ]_g: gi2 2P'

where we write i : id - Fw(n) for the unit of the monad on V+ (omitting the forgetful

functor). That is: we apply Pi, not to y E X, but rather to 7y, the corresponding generator

of F w(n)X; we apply P to 'q2 x, a generator of F , F ")X; the bracket is taken in the(1)(0 )

outermost free construction in Bw (o)X :- F W(n)F W(n) F W(n)X.

With this notation in hand, the map 3 is induced by the assignment

[Pis ... 1PIl x - P p0) - -- Pl _1)x (if n = 0),

xAI .. Ai.] i--+ xAi,(-l).. AiS_,(I) AiS(o) (if rn > 1).

Before making calculations, we recall the formulae of [20, 81 for the Lie algebra homotopy

operations discussed in 5.5. Let Shpq be the set of (p, q)-shuffles, that is, pairs (oz, 3) where

a = (ap-1,... , ao) and 0 = (1q-1, ... ,!o) are disjoint monotonically decreasing sequences

that together partition the set {,... , p + q - 1}. Let sa denote the iterated degeneracy

operator s, 1 .-. so. Finally, let Sh ;2 denote the subset of Shii consisting of those shuffles

(a, /) E Shii such that #i_1 = 2i - 1. The formulae of [20, 8], for z E ZKp(X) and

121

W C ZKq(X) cycles representing classes , E HC MX, are as follows:

[z, U] is represented by [ s(jz), sa(JW)](-) C Qu(n)B+qX;(a,I)EShpq

.Aj is represented by E3 [s,(3z), Sa(JZ)](_ E QU(n)B+ 7)X, (0 1).

Then z may be written as a sum of terms of the form xAj 1 (p_) ... A (o), and

Lemma 9.7. If (a, 3) G Shpq, then s,(xAi, (p,) - Aj-(o)) = xi(a_) - - -x( 1ao))

We will also need the following consequence of the simplicial identities:

Lemma 9.8. Choose i > 1 and a = (ap-1, . . . , ao) with ap-1 > - > ao > 0.

(1) If neither i - 1 nor i - 2 appear in a, then di_1s, = saldi, for some &' and i'.

(2) If exactly one of i - 1 and i - 2 appears in a, then di- 1s, does not depend on which

of i - 1 and i - 2 appeared in a.

Proposition 9.9. The rZ (n) bracket H X0 H X H vanishes except when

p = q = 0. The Lie algebra structure on He( )X is induced by that on X: if z, w c X

represent T, UT HC X then [T,7] is represented by the cycle [x, y] C ZC0 (QU(n)BW(n)X).

This theorem shows that HYX is trivial in positive dimensions as a Lie algebra, but

nonetheless, the restriction need not be trivial (c.f. Propositions 9.11 and 9.12).

Proof. We will give the proof for n > 1, but it works the same way for n = 0. In fact, when

n = 0 we can ignore all discussion of top and non-top operations.

Use the abbreviation B := QU(n)BW(n)X E sZ (n). Then B is almost free on the subspaces

V = F () - F ()X. Choose representatives z E ZKU n)X and w C ZKq n)X. For any

(a,3) C Shpq, the elements sos,8(3z) and sos(3w) of Bp+q+i both lie in Vp+q+i, and it is

only a minor abuse of notation to define:

a := E [sos (Jz), sosal(3w)](o) C Cp+q+ 1.(a,O)EShpq

What we mean here is that the bracket of the elements sos,3(3z) and sos((3w) of

Fw(n) . w(n)x CW(n)FW(n) 'vV F(n) X -B~V+q+1 = F O) ... F0 X C F_ nFj" ... F0 X = W(n)X

is taken in the free construction F w(n).(0)

122

Using the simplicial identity doso = id, we have doa = E[s,(jz), sa(jw)](-1), the rep-

resentative given for [z, ]. Moreover, we will find that dia = 0 for i > 0, except when

p = q = 0, in which case da = [x, y]. Thus, in either case, a is the required homotopy in

CIB.

Using the simplicial identity diso = id, we have da = Z[sa(jz), s,(3w)](o). Now for

every pair (a, /) indexing this sum, unless p = q = 0, one of a or #, say /, will contain

0. Then by Lemma 9.7, every summand in s,(jz) is in the image of some non-top Ajio),

and as [xAi, y] = 0 whenever Ai is not a top operation, the entire expression vanishes in the

construction F w(n)

What remains is to show that dia = 0 for 2 2:

dia = t [sodi -1s,8(z), sodi -Isa (3w)] (0).

For this, we will define an involution pi of the set Shpq indexing the sum, for 2 < i < p+q+ 1.

If a and / do not each contain exactly one of i - 1 and i -2, then pi fixes (a, /). Otherwise,

pi interchanges the positions of i - 1 and i - 2 in (a, /). To avoid confusion, we note that

Pp+q+1 is the identity, as neither a nor / ever contain p + q.

If (a, /3) is a fixed point of pi, then one of a and 8, say a, contains neither of i and i - 1.

Then by Lemma 9.8(1), di-is (w) = s'di, (jw) = 0, as jw E ZN; B. Thus, the summands

corresponding to fixed points vanish. On the other hand, given a shuffle (a,/3) which is

not fixed by pi, Lemma 9.8(2) shows that the summand corresponding to (a, /) equals the

summand corresponding to pi(a, /), so these two summands cancel with each other. l

In order to state our calculation of Ao of H*j0OX for X E W(0), define

adm+ (A, t) := {I I a non-empty J-admissible sequence with TM(I) 5 t}.

Lemma 9.10. There is an injective function Tt : adm+ (A, t) -+ adm+ (A, t) given by

I = ir..,i) -4 (t + nI+ t, ij,...,Iii).

Proof. This is indeed a well defined injective endomorphism of the set adm+(A, t), in that

it preserves admissibility and the condition 7iT(I) t. The claim about TE(I) holds by

definition. For 6-admissibility, as 7if (I) < t,

it < f - 1 + ie_1 + + i- + t

which implies the (strict) inequality 2ij < f + ij + --- + ii + t.

123

Proposition 9.11. Suppose that n > 0, z E (ZKs+l X) and 1 < k < s,+,, so that

zAk is defined. Then zAk = 0 unless n = 0 and k = 1.

When n = 0, k = 1 and sl 1, Al may be defined at the level of the Koszul complex as

follows. The generic cycle z E (ZKUjO)X)t may be written as a sum

z = Ej 6gxj, with xj E Xti and I C adm+( A,tj ) of length s1.

Then -'Aj is represented by the cycle

Ej6vt.Ijxj E (ZKU(O)X)2t+1.

Proof. We will first prepare for the calculation of A, in case n = 0, abbreviating s, to s. Note

that each Tt, appends the same integer, t, to Ij. Write e for the proposed representative

Z3 S(,t, )xi of 4Aj. Our first claim is that e = Ptso(jz), since

[pks+1 - pki I_ -- [pt [ ph., .. phi ] .

j, K (TtjI,) jHI,

The first of these two sums a priori contains more terms. However, the extra terms all vanish,

by the unstableness condition. More precisely: if (k,+l,... , ki) A TI and k,+ 1 # t, then

S(k,+1, .... , ki) > tj, so that [pks+ I .. jpki] xj = 0. To understand this observation, as 6-

Adem relations cannot increase 7 T (Lemma 9.3), we may reduce to the case where (k, . . . , kj)

is already 6-admissible, t # k,+1 , and (k,+, k,) - (t, k,+1 + k, - t), where::

7E(ks+1, ... , ki) > m(ks+1, ks) - (ks 1 + 1) - - - - - (ki + 1)

> 7T(t, k,+1 + ks t) - (ks-1 +1) ----- (ki + 1)

> 2t - (ks+1 + --- + k, + s)

= 2t -(t +is + k-+i1+s) =tj.

where: the two non-strict inequalities are by definition of 7i; the strict inequality follows

from Lemma 9.3; the first equation holds as A is graded by the sum of the indices; and the

second equation holds as t is the dimension of 6}*xj.

With this in hand, we return the general case, 1 < k 0, our goal being

to produce a nullhomotopy, except when n = 0 and k = 1, when we need a homotopy to

PO)so(jz). We proceed as in the previous proof, defining

a := 2 [sos(jz), sosa(z)](o) C Cp+k+1I sn..,2 s1 .(aSg)E.h

124

Then doa is the representative for TAk, and d1 a = 0 as in the previous proof (and there is

no analogue here of the special case p = q = 0). Now consider the same involutions pi as

in the previous proof, now acting on Shkk. When 2 < i < 2k, pi preserves Shi. When

2k < i < p + k + 1, pi is the identity, so preserves Shk trivially. Thus, dia = 0 for all

2 < i < p + k + 1 with i $ 2k, as the summands corresponding to fixed points vanish, and

the cancellations still all occur within the smaller sum

dia = E [sodil1s#(Jz), sodis,(jz)](o).(a,3)EShtd

To address the question of d2k, we define an alternative involution P2k of Shkk as follows.

If a and / do not each contain exactly one of 2k - 2 and 2k - 1, then ;2k fixes (a, /).

Otherwise, we define P2k(a,/) := P2k(3, a), which is to say that ;2k swaps everything but

2k - 2 and 2k - 1.

Now the summands in this formula exhibit a symmetry not present in the previous proof:

z is repeated. This symmetry, along with Lemma 9.8(2), shows that all the summands

corresponding to shuffles not fixed by p2k cancel out. When k > 1, the fixed points of P2k

are only those shuffles in which one of a and / contains neither 2k - 2 nor 2k - 1, and

the corresponding summands vanish, by 9.8(1), as in previous arguments. When k = 1,

however, ;2k has an extra fixed point, the shuffle ((0), (1)), which fails to differ from its

image under P2k. In this case, then:

d2 a [sodsI(z), sodiso (z)](o)

= [so(Jz), so(Jz)](o)

0, if n > 1,

(0) so(Jz), if n = 0.

That is, if n > 1, this self-bracket vanishes (an object of W(n) for n > 1 is a Lie algebra),

while if n = 0, the self-bracket is equal to the top P-operation, in this case Pt.

In sum, we have shown that doa = 0 represents Ai, and that dia = 0 whenever 1 1, and z E (ZK X),. where not all of

Sn,... , s, equal zero. If sn+1 = 0 then Ao is represented by zAn E X2 ,..,21

Suppose instead that sn+1 > 0, and consider a cycle

z = 2j Sq3 x E (ZK X) S

125

for various xj c X and Sq-admissible sequences Ij = (ij,S.+ 1 ... ij,1). Suppose further that

for each summation index j, xjAi_ 1 = 0 whenever i > i3 ,1. Then z AO = 0.

Proof. Write p := sn+1. The same homotopy a as in the previous cases shows that AO

is represented by (z) j = zAS,(o) when p > 0, and by zAsn E X when p = 0, so that

we may restrict to the case p > 0. Then, zAs8 ,o) is the image of the following element of

ZFp+1 N-QU(n)BU(n)X:

E = xi [Aki_1 - -Ak,_1-1 Ak,-1Asn]

j,KIl

X3 [Aki-1 ' '. Ak,_1-1jA--Aa-j, K- Ij (c,,O) (s,,+1,kp)

where the second equation holds by the Koszul duality of the A-algebra and the homogeneous

Steenrod algebra. As homogeneous Sq-Adem relations move Sq-inadmissible sequences to-

wards Sq-admissibility, when p > 2 we have ki ij,1 in each summand, and when p = 1 we

have / ij,i in each summand.

Dualizing Priddy's work, namely [47, Proof of Theorem 5.3], gives a sequence of homo-

topies which move this cycle into FpN . Indeed, given an expression

e = y [AgI_11- - -*Ag,_%1|Ag,_j_1Ag,_1iAg,+1-1 ... Ag,-1] C Fp+1Ng,

(with the composite Ag,_ 1iAg,_1 A-admissible), define:

IF (e) : [A 1_1 I ... IAg,-i Ag r - I I ...+Agp -] , if (gp+, . ,gr) is Sq-admissible;

0, otherwise.

If we further define F to be zero on FpNg, then F : F--+ Fp+1N +1 may be used as

a chain homotopy to compress E C ZFp+1 N into ZFpN:

(id + dF)UE stabilizes to an element of ZFN as u -- + oc.

A

As we repeatedly apply (id + dF) to this e, because al > b 1 whenever (b 2 , bi) -+ (a2 , a,), the

very leftmost A-operation in any of the expressions that appear is Am-1 for some m > g1,

and every term in (id + dF)ue c ZFpNg will be of the form yAm- I[ for some m > g1.

Applying these observations in the very specific circumstances of this proposition, along

with the earlier observation that in the sum defining the cycle E we always have ki ij,1

(or 3 ij,1 if 1), one derives that (id + dF)uE = 0, so that E is nullhomotopic. El

126

Chapter 10

Operations on second quadrant

homotopy spectral sequences

In this chapter we will produce various external operations on second quadrant homotopy

spectral sequences. That is, for X E csV, we will produce operations from [ErX] to each of

[Er S2X] and [ErA2X]

This approach leaves open a number of possibilities. If X E csWom, then the structure

map p : S2X -+ X induces a spectral sequence map [ES 2X] -+ [ErX], and so the

external operations induce internal operations on [ErX]. If X is a Lie algebra we may apply

the analogous technique [,] : A2 X -+ X. In 11, we will use these external operations in

another way to produce operations on the BKSS of a commutative algebra or Lie algebra -

the construction will involve a shift in filtration, which is conceivable given that Radulescu-

Banu's resolution is a resolution by GEMs.

A number of authors have written about spectral sequence operations in a variety of

settings. Singer's work [53] on first quadrant cohomology spectral sequences will be used

extensively in 13.1, and has been extended by Turner [56]. Perhaps the closest recent

examples are due to Hackney [37] and [36], who works out the operations available on the

homotopy spectral sequence of a cosimplicial E,,- or En-space respectively, using Bousfield

and Kan's universal examples [9]. We will be working with cosimplicial simplicial vector

spaces, and so we are able to develop a direct approach, mirroring Dwyer's work in second

quadrant cohomotopy spectral sequences [25].

Dwyer's work makes an interesting point of comparison with ours. In both cases: prod-

ucts, Steenrod operations and higher divided powers (as in [25] and 5.4) are produced on

the spectral sequence; one set of operations is not present in the target; the other set of

operations is present in the target, but the unstableness conditions on the target and on E2

do not agree; and differentials are constructed between the two varieties to simultaneously

127

rectify these disparities. Between Dwyer's theory and the theory presented here, the roles

of the two types of operations are interchanged.

10.1. Operations with indeterminacy

On pages higher than the E2-page, some of the 'operations' [ErX] -+ [ErS2 X] that we

would like to use will in fact fail to be well defined, and in this section we will introduce the

language which we will use in such situations.

We will make use of the notion of a (potentially) multi-valued function f : D -- C,

which is just a relation f C D x C such that for all x c D there exists some y C C for which

(x, y) E f. We may drop the modifier potentially, with the understanding that we do not

insist that a multi-valued function fail to be a function. If a multi-valued function turns out

to be an actual function, we will call it single-valued. For x E D, the set of values of f(x)

is {y E C I (x, y) C f}.

In all of our examples, D and C will be vector spaces. A multi-valued function f :D -

C has linear indeterminacy if it is essentially a map D -- + C/I for some subspace I of C.

That is if there exists a subspace I of C such that for all x C D, the set of values of f(x)

is a coset of I in C. Such a function is linear if f(x + y) is the sum of the cosets f(x) and

f(y) for all x, y C D. We will refer to multi-valued functions with both properties as linear

with linear determinacy. Almost all of the multi-valued functions we encounter are linear

with linear determinacy (all of the exceptions are operations on pages E0 or El, or are top

6-operations, c.f. 10.5 and 11.3).

In this chapter, multi-valued functions will arise in two ways. An operation [ErV] -+

[ES 2 V] with indeterminacy vanishing by Er' will be an actual function

[ErV -+ [ErS 2V]/[Br,rS 2 V]

where [B,,r S2 V] C [ErS 2 V] is the subgroup consisting of those elements which survive to

[ES 2 V] and represent zero there. We view such operations as linear multi-valued functions

[ErV] -+ [ES2 V],

and the external Steenrod operations that we will define in 10.4 will be examples. On the

other hand, we will define in 10.5 external 6-operations which will sometimes be multi-

valued, and will almost always be linear with linear indeterminacy. We do not expect a

vanishing result for the indeterminacy of these external 6-operations, as we expect them to

support a differential.

128

10.2. Maps of mixed simplicial vector spaces

For mixed simplicial vector spaces X, Y E csV, we will write C(XOY) for the double complex

associated with the levelwise tensor product of C(X 0 Y), so that C(X 0 Y)' = Xt Yt.

We will write C(X Ov Y) for the double complex with C(X Ov Y)' = ]t'+t"=t Xs, 0 Xs,.

The following vector space maps are given by prolonging Dk, V, Vk and Ok wherever these

maps are defined, and by zero elsewhere:

Dk : (CX ( CY)s+k -+ C(X O Y)s (zero unless 0 < k < s)

V : C(X o0 Y)s -+ C(X 0 Y)s (no condition)

Vk : C(X 0 Y) +k -+ C(X 0 Y) (zero unless 0 < k < t)

Ok C(X 0Y Y)tsk -- C(X 0 Y) s (zero unless k = t > 0)

We have just committed to regarding Vk as zero where it is not defined. This is certainly

not a natural convention, and it has somewhat untidy results, for instance:

Lemma 10.1. Suppose that z E C(X 0 Y)'. Then

(dVk + Vkd)z = ((1 + w)Vk+1 + Ok+1)z

whenever k > 0 and t does not equal either of 2k and 2k + 1.

As discussed earlier, we will write T for any symmetry isomorphism, write "wG" as

shorthand for the function TGT, and whenever we write wGH, we will mean (wG)H. We

will also use the notation

X2 -P S 2 X and X02 2

for the projection onto coinvariants and further onto the exterior quotient. Until 10.5, the

operations that we will produce into each of [ES2 X] and [ErA2 X] will be essentially the

same.

10.3. An external spectral sequence pairing etxt

The easiest and most standard of our constructions is that of an external product, using the

chain-level formula

x 0 y -+ pVD(x o y).

Both V and D0 are chain maps, and filtrations add under VD0 , and thus:

129

Proposition 10.2. The map pVD0 (x 0 y) : CX 0 CX -+ CX induces a pairing

[ext : [ErX]9 9 [ErX|15 -- [ErS2X]+s

for each r, satisfying the Leibniz formula. For r > 2, this map descends to the symmetric

quotient S2 [ErX]. Under the identifications [E2 X]t ' ir'wiX and [E2 S2 X]' qr7r 'S 2 X,

[text corresponds to the composite

S2 7rh* '* X*"4 7rj*(S27r'X) > )r*rvS2X.

10.4. External spectral sequence operations Sqe

Consider the chain-level map:

SQi's : x - pV(Ds-i(x 9 x) + Ds~i+1 (x 0 dx)).

We will use these maps to define external Steenrod operations Sqext, the behaviour of which

is rather different on El than on later pages. Thus, we will state two separate Propositions

that we will prove together.

Proposition 10.3. Suppose that r > 2. The chain level operation SQis defines a linear

operation with indeterminacy vanishing by E2 - 2 :

Sqext : [EXI -+ [ErS2X1]t2 .

Now suppose that x E [ErX]t. Sq',,x = 0 unless min{t, r} i < s, and this vanish-

ing occurs without indeterminacy. In any case, Sqextx survives to [E2 rlS2X]sti, and the

following equation holds in [E2r-1 S2 X] 2?i21 (without indeterminacy):

d2r-I(Sqextx) = Sqxt- 1 (drX).

The top operation Sq xtx is equal to the product-square Pext(x 0 x), and in particular

has no indeterminacy. As for the only potentially non-zero Sqext operation:

oX squarilngSq0 t : [ErX] [ErS2 X1]' is induced by X su n 2X.

At E2 , there is no indeterminacy, and the operation Sqext corresponds to the composite:

7r1rSX s t(+iS2(7rX) h ( h+irtS2X.

The condition min{t,r} < i < s may be replaced with min{t + 1,r} < i < s after

composing with [ES2X]tz + [E k2 X]s.

130

Proposition 10.4. At E1 , the chain level operation SQi's defines an operation

Sq'.t : [ErX]" -- [E,S2 X]itZ

which commutes with the differential d1 . Suppose that x C [E1X]'. The top operation Sqs ex

need not equal the product-square btext(X 9 x) on E1 , and Sq'+1 x need not vanish, instead

equalling btext (x 0 dix) on E1 . The operations have no indeterminacy and need not be linear.

For i > s + 1, Sqixt = 0. Sqlx is zero whenever t > 1. Sq0x = 0 for all t.

Proof of Propositions 10.3 and 10.4. Choose a representative x E [ZrX]' of the class of

interest. We readily check that SQi''(x) has filtration at least s + i:

filt(pVDS-i(x 0 X) s + s - (s - i) - s + i,

filt(pVDS-i+(x 9 dx) s + (s + r) - (s - i + 1) = s + i + (r - 1).

Thus, we may view SQi''(x) as an element of [ZoS2X]'t . A straightforward calculation

shows that

d(SQi''(x)) = pVDS-+1(dx 0 dx) - SQi+r-ls+r(dx),

and as x C [ZrX ]:

filt(d(SQi''(x))) > (s + r) + (s + r) - (s - i + 1) = (s + i) + (2r - 1),

so that SQi''(x) E [Z2r-lS2X]rt+. This demonstrates the survival property, along with the

formula commuting the Sq'xt with spectral sequence differentials.

The next step is to examine the non-linearity of the operation SQi's, which we do using

formulae analogous to [53, (1.111) and (1.112)]. That is, for x, x' E [ZrX]', one calculates

NL(x, x') := SQi''(x) + SQi''(x') + SQi' 8(x + x')

= dpV [Ds-i+2(x 0 dx') + DS-i+1(x' O x)] + pVDS-i+2 (dx 0 dx').

The first two terms of NL(x, x') are the boundaries of chains in filtrations satisfying

filt(pVDs-i+2 (x 0 dx')) s + s + r - (s - i + 2) = (s + i - r + 1) + 2(r - 2) + 1,

filt(pVDs-i+ '/ x x)) > s + s - (s - i + 1) = (s + i - r + 1) + (r - 2),

so that they vanish in [ErX1't2 whenever r > 2. Moreover

filt(pVDs-i+2(dx o dx')) 2(s + r) - (s - i + 2) = s + i + 2(r - 1),

131

so that the final term also vanishes in [EX'+' when r > 2. When r > 2, this proves that

whatever indeterminacy these operations are subject to is linear, and that the operations

themselves are linear.

To examine the indeterminacy, as a representative of a class in [EX]', x is only deter-

mined up to boundaries of y E [Z,_ 1 X]-$2+ and elements of [Er- 1X]'+'. The latter are

irrelevant, as their effect on the value of SQ2'8 is restricted to high filtration. The boundaries

dy are more problematic, but if we define

BC(x, y) :=pV[D'-- 1(y 0 y) + D8 -(y 0 dy) + D-'+ 1(dy O x)]

then this chain has boundary

d(BC(x, y)) = pV[DS-l(dy 0 y + y 0 dy) + Ds-i(dy 0 dy) + Ds-i+1 (dy 0 dx)]

+ pV[O + D''~(y 0 dy + dy 0 y) + Ds-(dy O x + x 0 dy)]

SpV [Ds-8 (dy O x + x 0 dy + dy 0 dy) + D'-'+1(dy 0 dx)]

= SQ '8 (x) - SQ2'8(x + dy).

That is, BC(x, y) is a bounding chain for this difference, and

filt(BC(x, y)) 2(s -r 1)- (s - i - 1) = (s + i) - (2r - 3),

so that Sqxtx has indeterminacy vanishing by [E2 r-2 2 X] t2 as claimed. When i = s,

this result may be improved to filt(BC(x, y)) > 2s - (r - 1), as in this case the lowest

filtration summand in fact vanishes - this is one explanation of why the top square has no

indeterminacy.

When i > s + 2, we have SQ2 (x) = 0, and even with i = s + 1:

SQs+1,s (x) = pVD0 (x 0 dx) E F2,+r,

so that Sq+ 1 x vanishes when r > 2, and Sq'+1 x = pext(x 0 dix) when r = 1, without

indeterminacy in both cases.

We must also check that Sq' tx vanishes (without indeterminacy) when i < min{t,r}.

For this we use the filtration preserving operations DELi to be defined in 10.5. Suppose

Proposition 10.7 states that

d(DELt-i+I(x)) + DELt-i+l(dx) = SQi's(x)

132

as long as 2 < t - i + 1 < t + 1 (which is satisfied whenever i < t). Moreover, if i < r, then

DELt-i+i(dx) E Fs+r C Fs+i+l and DELt-i+i(x) E F',

so that this equation states that SQ''(x) - 0 in [EX]t", without indeterminacy.

For the statement about the top operation, one calculates that

SQ','(x) - pVD(x 0 x) = pVD 1(x 0 dx) c F

which exceeds filtration 2s when r > 2.

For the statement about Sqextx when t = 0, using the specialness assumption:

SQOs'(x) - pVDS(x 0 x) = pVDs+1(x 0 dx) E FS+1,

so that (using the assumption that {Dk} is special):

SQO,'(x) = pVD8 (x' 0 x')

= P(0o + (1+w)Vo)(x OV xt)

(mod Fs+l)

(Xt ov XS" E C(X ov X)St)

- pbOO(xt (V x) C C(X 0g2 X)t.

The statements about [ErA 2X]ti2 follow similarly, replacing DEL with LAMi. 0

10.5. External spectral sequence operations &gxt

For any k (positive or otherwise) write Dk : (C(X) 0 C(Y))i -+ C(X 0 Y)ik for the map:

Dr(Z)= VowD,(z).

Lemma 10.5. If x C FSC,(X) andy C F8 'Cn'(X), then

Dk(x 0 y) E Fmax{s,s'}Cn+n'-k (X 0 X).

Proof. We may assume that x and y are each homogeneous, with x E Xt" and y c Y . As

{Dk} is special, D3(x 0 y) = 0 unless 3 < min{s, s'}, in which case

filt (DO (x 0 y)) s + s' - # > s + s' - min{s, s'} = max{s, s'}. El

133

Lemma 10.6. For all k (positive or otherwise) the equation:

(dDk + Dkd)(z) = ((1 + W)Dk+1 + VwD-k-l)(Z)

holds when z G (CX® CX)N with N > 2(k + 1). When N = 2(k + 1),

(dDk + Dkd)(z) = ((1 + w)Dk+1 + VwD-k-1 + Ea awQa+Da-k-)(z).

Proof. We may assume that z is homogeneous, z C (CX ® CX)s with N = t - s > 2(k + 1).

Choose a and 0 such that a - 3 = k. Then (wcDO(z)) E C(X 0, Y) .

We will need to apply Lemma 10.1 to calculate, for a - : =

(dV, + V d)(w'Dfl(z)) = ((1 + w)Va+l + +1)(w'D3(z)),

but Lemma 10.1 does not apply when t = 2a + e for e C {0, 1}. Fortunately, in that case

DO(z) is zero, so the equation holds by default: after all, if t = 2a+e, our assumed inequality

on N implies:t-e -k>t-e t-s-2 _ s+ 2-e s

2 ~ 2 2 2 2

After these observations and under our conventions on the Vc and DO, all but one of

the following manipulations is totally formal:

(dDk + Dkd)(z) := E (dVwQDfl + VwaDd) (z)a-3=k

= 1: ((dVc, + Vad)waDO + Vcwca(dDO + D3d)) (z)a-/3=k

= >3((1+W)Va+1 + Oa+1)wcDI(z) + 1: Vaw"(1+w)D'- 1 (z)a-o=k,a>O a-o=k

= >3((1+w)Va + 1)w"~1D3(z) + E VLwa(1+w)D3 (z)a-3=k+1, a >1 a-O=k+1

Using the identity (1 + w)Vo + #o = V for the first equation, and the observation that

(1 + w)FwG + F(1 + w)G = (1 + w)(FG) for the second (with F = Va and G = 'D#):

(dk + IDkd)(z) - VwD-k-l(z) = (((1 + W)Va + 0,)waDI3 + V,(1 + w)wQD,3) (z)a-3=k+1

((1 +w)(Vow"D 3 ) + W a+1D/3) (z)a-O=k+1

= (1 + w)Dk+1(Z) + > aWa+Da-k-(z)

134

When the strict inequality t -s > 2(k + 1) holds, due to the application of 0, each summand

4awa+1Da-(z) is zero unless t = 2a, but in that case, s = 2a - N < 2(a - k - 1), and

then Da-k-l(z) vanishes as {Dk} is special. 11

We will be able to define (sometimes multi-valued) operations (for r > 0):

Jext [ErX]' -+ [ErS2 X]'+i for 2 < i max{n, t - (r -1);

Aext : [ErX]s - [ErA 2X],+i for 1 < i < max{n,t - (r - )};

using the chain-level maps DELi : C*X -- + CS2 X and LAMi : C*X -+ C*A 2 X:

DELi(x) := p (DI1hi (x 0 x) + D,_i_ 1(dx 0 x));

LAMi (x) p'(D,-i(x 0 x) + D-i-1 (dx 0 x));

where we write n := t - s in each formula. Except when i < 2, we can work just with the

DELi, as in 5.2. Lemma 10.5 shows immediately that these maps preserve filtration, in the

sense that DELi(x) E FsCn+i(X 0 X) whenever x E F8 CnX. Moving forward we will need

a formula for the boundary of DELi(x):

Proposition 10.7. For 2 < i < t + 1 and x E [ZOX],:

SQt-i+1,s (W,d(DELi(x)) + DELi(dx) = SQt-+"s(x) = pVD0 (x 0 dx),

0,

if n + 1 2(n-i+1) and Jdx(9xJ=2n-1 >2(n-i-1+1) wheni>2.

Note that the first inequality fails when i = 1, which will explain the lack of 3 ixt. We can

work around this difficulty when defining A'xt (the final step of this proof). Returning to

DELi for i > 2:

d(DELi(x)) + DELi(dx) = pd(Dn-i(x 0 x) + Dn-i-1(dx 0 x)) + pDn I(dx 0 dx)

= P{dDn-i(x 0 x)} + p{dDn-i-1(dx 0 x)) + Dn-i_1(d(dx 0 x))}

= P{Dn-id(x 0 x) + (1 + w)Dn-i+ 1 (x O x) + VwD'-n- (x x)

+ p{(1 + w)lDlY-i(dx O x) + V7wD'~(dx 0 x) ,

135

where we used braces to indicate the two applications of Lemma 10.6. Everything cancels

except for pV(D'-'- 1 (x 0 x) + D'-n(x 0 dx)) which equals SQt'-'+,s(x). We have studied

this expression above, explaining the three cases.

If i = 1, Lemma 10.6 yields an extra term, and if we write x as the sum ET xf- of its

homogeneous parts, as {Dk} is special, this term is:

p(>i 4pwQ+1D"-l(x 0 x)) P(ZT xT- 0 x~~) E S2X.

Although this term need not vanish, its image in A2 X certainly does, so that LAM, satisfies

the desired equation. D

Suppose now that x E [EX]t. In light of the above calculation, when n < i < t + 1,

the purpose of jext(x) will be to support a dt-j+1-differential to Sqt- +1 (x). Thus, we would

not expect to be able to define 6ixt(x) when t - i +I1 < r; indeed, the following result will

construct J5xt(x) whenever i < t - (r - 1).

Moreover, Sqt-'+1(x) has indeterminacy vanishing by [E2(r-lyS2XJit , so we should

expect that whenever t - i + 1 < 2(r - 1), 6 xt(x) will be multi-valued, but that the set of

values for c5xt(x) will map onto the set of values for Sqt-+l(x) under dt-j+. We are not

saying that we expect the indeterminacy of J xt(x) to vanish by a certain page, but rather

that we expect the multiple values of 6 ext(x) to all fail to be permanent cycles together.

Note that when r < 2, there is no indeterminacy whatsoever in either set of operations.

Proposition 10.8. Suppose that r > 0. The chain-level map DELi produces a multi-valued

operation

6i: [EX] -+ [ ErS2X]8i defined when 2 1 and i < t then this function is linear with linear indeterminacy. This operation is

single-valued whenever 2 iS2(Xs

Suppose that r > 1 and x E [ErX]s, and suppose that 39xt(x) is defined. Then 65xt(drx)

is defined and

Sq-+1(x), if i > t -s and r = t - i + 1;

drc5ixt(x) + 6'xt(drx) = ext (X dx), if i = t - s, s = 0 and r > 2;

0, otherwise.

136

If 2 t - s and r = t - i + 1 the set of

values of the left hand side coincides with the set of values of the right hand side. Otherwise,

this equation holds modulo the indeterminacy of the left hand side.

For r > 1, the only potentially non-linear operations are

ie~t [E1 X]s --- + [E1X]'t and 3 ext [ErX]o + [EX]t. (10.1)

They have no indeterminacy and satisfy 6'xt(x + y) =e t(x) + jext(y) + bext(X 0 y).

The same conclusions hold for LAMj, producing operations A'xt, and the inequality 2 0, then pext(x 0 x) = 0 E [E.S2X]

because of the differential

dex t Sqextx = /text(X 0 X).

That is, although Pext (x 0 x) need not equal zero on the E2-page, the E,-page mimics an

exterior algebra in positive dimension. The fact that this top Steenrod operation has no

indeterminacy (c.f. Proposition 10.3) should be compared with the fact that 6eits+x has no

indeterminacy.

Proof. Suppose that x E [ZrX]'. Proposition 10.7 shows that dDELi(x) E Fs+rC(S2 X) as

long as i < max{n, t - (r - 1)}, so that DELi(x) E [ZrX]'+i. Proposition 10.8 then provides

the formula for dr,6xt(x) + fext(drx) (modulo whatever indeterminacy we find).

Let us begin with the operations 6 ext : [E1X] -+ [EIS 2X]' j. Due to the assumption

that {Dk} is special, for any x, y E F8 CX, Dk(X y) Vk VJkD(xy) modulo F2S+ X, and

due to Lemma 10.5, we can ignore the horizontal component of the differential dx appearing

in the definition of DELi(x). The resulting operations have leading term which is almost

identical to the definition of the operations 6ext of 5.2, which we already understand well.

The only difference is the w operator that appears, but this does not affect the resulting

operation, by [26, Lemma 4.1]. This calculation at E1 also demonstrates the expression

for 6txt(x + y) for the operations in (10.1): such an equation is known from 5.2, and as

[ErX] C [EIX] for r > 1, this equation persists for all of the operations of (10.1).

Next, suppose that 2 < i < t, and that X, X' E [ZrX], and define:

NL'(x, x') := DEL (x) + DEL '(x') + DEL2 (x + x')

137

By a calculation using Lemma 10.6 (similar to the calculation of dH that follows shortly):

d(NL'(x, x')) = dp[D,-i_1(x O x') + Dn-i- 2 (x 0 dx')]

+ p[Dn-i- 2 (dx 0 dx') + VwD'-'(x 0 x') + VwDi-n+1 (x 9 dx')]

The terms on the first line are zero in [ZrX]'+i, as they are the boundaries of chains of

filtration at least s. The three remaining terms lie in filtration exceeding s as long as i < t.

Thus, whatever indeterminacy these operations are subject to is linear, and the operations

themselves are linear.

We will now examine the extent to which DELi induces a well defined operation with

domain [ErX]' for r > 1. We may assume that s > 0, as the operations on El were shown

earlier to be well defined, and [ErX]2 C [E1X]O for r > 1. This implies that t - i >

1 whenever i = n. To examine the indeterminacy in 6ixtx is to examine the difference

DELi(x) - DELi(x + dy) for y c [Zr-iX]'r+2', and by Lemma 10.6, we have the following

three equations:

dDn-i+1(y y) = IDn-i+1d(y 0 y) + (1+w)IDn-i+2 (y 0 y) + V0D-(n-i+2)(Y y);

dDn-i(dy 0 y) = Dn-i(dy 0 dy) + (1+w)Dn-i+ 1 (dy y) + VD-(n-i+)(dy 0 y);

dDn-i_1(x 0 dy) = Dni 0(dx dy) + (1+w)Dn-i(x 0 dy) + VwD~(n-i (x 0 dy).

(As in the proof of Proposition 10.7, there are extra terms which appear when i = 1, but

they are annihilated by the application of p'.) We define the following chain:

H(x, y) := P(Dn-i-i(x 0 dy) + Dn-i(dy 0 y) + Dn-i+i(y 0 y)),

and note, by Lemma 10.5, that H(x, y) E Fs-r+Cn+i+1 (S2X). The three equations above

show that

d(H(x, y)) = DELi(x) - DELi(x + dy) + Ti + T2 + T3 ,

where

T:= pV(Di-n- 2 (y 0 y)) E FS+(t-i)-2 (r- 2) equals zero when i < n + 1;

T2 :=pV(D~-1nl(y 0 dy)) C Fs+(t-)-(r-2 ) equals zero when i < n;

T3 pV(D-n(dy 0 x)) C Fs+(t-i) equals zero when i < n - 1.

As t - i > 1, T3 E FS+i can be ignored. As we have supposed that the operation 6Y't can be

defined on x, we must have either i < n, in which case T2 = 0, or i < t - (r - 1), in which

case T2 C Fs+1 can be ignored. T3 is assured either to vanish or to lie in Fs+l exactly when

138

2 < i < max{n+1I, t+1-2(r-1)}, in which case, we have shown that oet x is single-valued.

In every case we may summarize the situation as follows. There is some H(x, y) C FS-r+l

such that

d(H(x, y)) = DELi(x) - DEL (x + dy) + BC(x, y),

where BC(x, y) : T1 + T2 + T3 is an example of the bounding chain appearing in the proof

of Propositions 10.3 and 10.4, so that

d(DELi(x)) - d(DELi(x + dy)) =SQt-i+s(x) _SQt-i+,s(x + dy) (mod FS+t-i+2),

and the set of values of 6 etx maps onto the set of values Sq-+ 1 x under dt-+j. El

Proposition 10.9. Suppose that X E (sV)'+, i.e. that X admits a coaugmentation from

some X~1 c sV. For 2 < i < t - s, the operations Jext : [E X]s -- [E* S2X]S++I agree

with the homotopy operations 6 xt : ,rt-s(X-1) - rt-s+i (S 2 (X- 1 )). Similarly, the external

pairing at S2 [E X] -- + [ ES 2 X] agrees with V : S2 r.(X-1 ) -+ (S2(X1)).

The same conclusions hold for the Ai for 1 < i < t - s.

Proof. We will only prove the statement about 5ixt, as the statement about products is

easier and more standard. We need to show that the following diagram commutes whenever

2 ZCn(S2X)do t do f

ZCn( X-l) > ZCn(S2(X -I))Zdp(Vdi(z&z))

We calculate

DELi(doz) := pn-i(doz 0 doz) + pDn-i_1(doz 0 d(doz))

= (doz 0 doz) + pDn-i- 1 (doz 0 0)

= pV-i(doz Ov doz) = do(pV-i (z 0 z)),

where we have used the assumption that {Dk} is special in both the second and third

equations, and d(doz) - 0 since z E ZC,(X- 1) is a (vertical) cycle, and do equalizes do and

d'. El

10.6. Internal operations on [EX] for X E csom

Suppose that X C csWom is a cosimplicial simplicial commutative non-unital F2 -algebra. In

this section we will define operations on [ETX] by direct application of the structure map

139

of X. Although these operations will not be used in rest of this thesis (they equal zero in

the case of interest to us), we hope they are of some independent interest.

We define operations:

i ([EX]t ] 4 [Eq2F"9omX]8 i *> [ErX]i) ,

Sq : ([EX]s Eq2F'e Xs ] ! * [ErX] j')

yu: ([ErX]s 0 [E,.X]tl AL'" [Erq2F4omX]s+t,,' >*+ [E,.X]5s'

with the Ji multi-valued functions, defined when 2 < i < max{n, t - (r - 1)}, and single-

valued whenever 2 < i < min{n + 1, t + 1 - 2(r - 1)}, and the Sq multi-valued functions

with indeterminacy vanishing by E2r-2, and which equal zero unless min{t, r} < j 5 s.

Numerous properties of these operations follow directly from the earlier results, namely

Propositions 10.2, 10.3, 10.4 10.8 and 10.9. In addition, we have

Proposition 10.10. The operations 6i : [E1X]l ---+ [EX i are (the restriction of) the

homotopy operations of 5.4 applied to the homotopy of the simplicial algebra Xs. Moreover,

for each s, 7r'X' is a graded commutative algebra (again, c.f. 5.4), and the operations yi and

Sq' on E 2 are the standard operations on the cohomotopy of the cosimplicial commutative

algebra 7r'X'. As such, the operations Sq make [E2X] is an unstable left module over the

homogeneous Steenrod algebra, and satisfy the evident unstableness condition and the Cartan

formula.

If x C [E1X]' and 2 < i < 2t (so that the 6 t* operation that follows is defined), then

JiSqix = 0 C [E1 X] t j. If also y E [E1X]i:, and 2 < i < t + t', then 8i(xy) = 0.

Proof of Proposition 10.10. Everything here is straightforward, and we will present the cal-

culation SjSq'x = 0 as an example. Suppose that x c [Z1X]' and 2 0, and for our current purpose we can assume that X E X,, so

that dyx = 0. Then Sqitx is represented by the image of Ds--i(x 9 x) + D~j+l(x o dhx)

under the composite

Ns+j(NtX 0 NtX) N+ ) Ns+jN2t(X 0., X) -L- Ns+jN2t(X) '_> Ns+jN2t+i(X),

and Proposition 5.3 states that the final 6-operation annihilates products of positive dimen-

sional classes, so that this composite is zero. El

140

As a final note here, suppose that X E csfie (or cstier). We may define operations:

A [ EX ]s -'L [Erq2FY'eXs ]ii _:_ [ ErXs ]i

Pj-1 : [[EX| s- [ Eq2F~ieXQs5 1,* ( EX ,~

({ [ErX 0 [ErX3|l L [ Erq2F'eieX+s| [14 [ErXt+')

with the Ai multi-valued functions, defined when 1 < i < max{n, t - (r - 1)} and single-

valued whenever 1 < i < min{rn + 1, t+ 1 - 2(r - 1)}. It should be possible to state versions

of all of the above results in this case. The author guesses that the operations Pk will form

an unstable left action of the P-algebra (the Steenrod algebra for commutative F2 -algebras,

as in 6.6) but has not worked out the details.

141

142

Chapter 11

Operations in the Bousfield-Kan

spectral sequence

In this chapter we will define operations on the BKSS for an object X E se whenever C

is any of the categories Wom, 2ie or Yier. We will always write X for Radulescu-Banu's

resolution of X c sC, the coaugmented cosimplicial simplicial object defined by

tX = (c(KeQec)s+lX)t.

11.1. An alternate definition of the Adams tower

We will now give an alternate definition of the Adams tower of 4.2, using the techniques of

[111, which is more suited for the definition of spectral sequence operations in our setting.

For Z E Vp+, the category of coaugmented cosimplicial vector spaces, Bousfield and

Kan write VZ for a "path-like construction" [11, 3.1] obtained by shifting Z down and

forgetting the oth coface and codegeneracy. That is, (VZ)S := (VZ)s'+, and:

((V Z) 8d (VZ)s+l) :=(Z8+1 d'+ Zs+2)

((V Z)" +_ (VZ) -) := Zs Z')

The unused coface do induces a map v : Z -4 VZ in V'+.

For Y E sV, the standard simplicial path fibration (c.f. [12, p. 82]) produces a con-

tractible simplicial vector space AY E sV by shifting down and restricting to a kernel:

AY, = ker (ds+ .. -di : Ys+1 -- Yo).

We forget the 0 th face and degeneracy as before, and this time, the unused face map do

143

induces a fibration A : AY -+ Y.

Each of these constructions can be prolonged to an endofunctor of (sC)A+, endofunctors

which are necessary for a key construction of Bousfield and Kan [11]. Define an endofunctor

R 1 of the category (sC)A+ of augmented cosimplicial objects in se, using the pullback (for

W E (sc) +):

R1 W - AVW

W V -- VW

Then one can form a tower in (sC)A+, (writing R: R1 o ... R1):

--- o R2W >- R1 W >ROW =:-W.

Restricting to augmentations, there is a tower of fiber sequences in se:

-. - > (R 2 W)- 1 - (R 1W)-l > (R0W)- 1 W-1

00 00 do

(R2W)0 (R1W)0 (ROW)0

Bousfield and Kan [11, 3.3 and 4.2] note that this tower equals the Adams tower {RX}

when W = X is Radulescu-Banu's resolution of X G sC. They also explicitly perform the

resulting identification of the El-page of the spectral sequence of this tower with NgirtW,

using iterates of the connecting map

7rt(W) = 7rt(VW ~l) "-""4 rti(RW)s~l

of the fiber sequence (R1 W)s-l --- Ws-1 -+ VWI-1, which has the property:

Proposition 11.1 [11, Proposition 5.2]. The following composite involving the connecting

map (c9.fnl induces (for each fixed t) an isomorphism of cochain complexes:

NgirtW C N'7rtVW 9-""# N-17rt_1(Rt1 W).

The inclusion in this theorem can be strict, as the subspace NCirtW of Cs7rtWs is defined

by the vanishing of the maps s0,... ,sS-1 : 7rtW8 -+ 7rtW- 1, while N-'7rt(VW)"- is

defined by the vanishing only of s,..., s -1: rtW -+ tW'~ 1 , by definition of VW.

If we declare the spectral sequence an object W E csC to be the spectral sequence of the

tower

t s . (R2W)- > (RW)-l (Ro W) - 1

then the spectral sequence of R'X maps to the spectral sequence of X, with a filtration shift,

144

via the map of towers:

- - > (R2R1X)- > (R1R 1X)-l-i > (R0R'X)- 1

- -- > (R3X)- : (R-2) (R1X)- 1 > (R0X)-l

That is, there are spectral sequence maps which at E1 are isomorphisms of the form

[E1R'X]t 2A4 [E ]+1 .

Under Bousfield and Kan's identification of E1 , this isomorphism is the inverse of the com-

posite of Proposition 11.1.

A reasonable goal is to create a natural factorization

q2F' X >

of the structure map of X through 6, as X is a GEM levelwise. This will be possible up to a

natural zig-zag, by a construction which uses the structure of Radulescu-Banu's resolution

specifically.

11.2. A modification of the functor R'

Not only does

(VX)S = (c(KeQeC)s+2 X)t E csC

have cosimplicial and simplicial structure maps, but there is a cosimplicial simplicial algebra

structure on the object VX obtained by omitting the leftmost replacement c:

(VX)S = ((KCQCc)S+2X ), E csC.

That is, we do not need the outermost cofibrant replacement in order to define the cosim-

plicial structure maps VX, as in passing from X to VX one discards do. There is a csc-map

e : VX -+ VX which is a weak equivalence in each cosimplicial level. Finally, the composite

v:=c 6o V: (x -- + Vx -> Vx)

is, in each cosimplicial degree, a fibration in sC since it is defined in cosimplicial degree s by

the formula

V= rq: c(KCQCc)s+ 2 X -+ KeQeC(KCQec)s+2X.

145

The object V has two key advantages: T is a fibration in each cosimplicial level, and VX

is a trivial object in sC (i.e. it is in the image of KC). This second property implies that VX

is an abelian group object in sC in each cosimplicial level, as every vector space is a group

object, and K2 is a right adjoint. In other words, since all the structure maps in VX are

trivial, they commute with vector space addition. We write add : VX x VX -+ VX for the

group operation. Under the identifications arising from Propositions 3.5 and 3.8, the map

add induces the expected abelian group and cogroup structures on H2VX and H*VX:

H$VX x HCVX -+ HTVX;

H*VX i H*VX <- HCVX.

The observation that T is a fibration leads us to define R X to be the strict fiber

0

There is a commuting diagram in csC (in which double-headed arrows denote maps which

are fibrations in sC in each cosimplicial level):

0 - AVX : -AVX

V V VI

pullbacks: RX El E-eq EiT 1-eq X

producing a zig-zag of Ei-equivalences between 1 X and R1 X. In each cosimplicial level,

each of the objects in the top row is contractible, yielding homotopy long exact sequences,

and the resulting connecting homomorphisms commute:

7rt (W1 X) "" 7rt+i (VX)zig-zagIt

srt(Ral) sE (rt+1 (V f)

so that there are isomorphisms of spectral sequences (starting from E1):

[- ET1 EI ]1 8 r 1- Erkr$q

146

11.3. Definition and properties of the BKSS operations

Whichever of the three categories of interest C we are working in, there is a factorization

R1X0induced by .

q2 F' > q2 F2 > V 00 VX

where the composite q2FeX -+ VX must vanish as it factors through the structure map

q2 FCVX -+ VX, which is zero since VX is a trivial object. We denote by

L : ~[Eq2 FeX]' - [E.X]j+'

the resulting map of spectral sequences. Using the isomorphisms

S2V L q2 F'emV, A 2 V f q 2F eV, S 2 V e q2FzierV,

and the various external spectral sequence operations from [ErV) to each of [ES2VI,

[ErA 2 V] and [ES2 V], we are now able to define numerous spectral sequence operations

on [EXI]' in each case. When e = Wom, we define:

( : [EX]s T [Eq2FX]s L + [EX]s+)

Cj-1 L+Sq : ([ErX] s'2i [Eq2FCX]-h -4 [EX]si)

pi: ([ EXT 0 [E.X]t, IL24 [ Erq2FCeX+; - [ ErXI )

with the 6i multi-valued functions, defined when 2 < i K max{n, t - (r - 1)}, and single-

valued whenever 2 < i < min{n + 1, t + 1 - 2(r - 1)}, and the Sq7 multi-valued functions

with indeterminacy vanishing by E2r- 2 , and which equal zero unless min{t, r} < j s + 1.

All of the functions that are defined on E2 are single-valued, so it makes sense to state

Proposition 11.2. When C = Wom, under the identification [E2 X]S H H* X, the

operations just defined coincide with the W(O)-cohomology operations defined in 8.

We will prove this result in 11.9. It implies that from the E2-page onward the operations

just defined have the properties cataloged in Propositions 8.2, 8.9 and 8.12 - the 6Y satisfy

the 3-Adem relations, the Sq' and p satisfy the properties of such operations on Lie algebra

cohomology, and there is a commutation relation between the 6 ' and the Sqh and p. These

relations persist to relations on the higher pages (modulo appropriate indeterminacy), but

evidently do not hold on E1 .

The following results follow from Propositions 10.2, 10.3, 10.4, 10.8 and 10.9 respectively,

147

for X E sMom and X E csWom its Radulescu-Banu resolution:

Corollary 11.3 (of Proposition 10.2). The pairing p satisfies the Leibniz formula. For

r > 2, [ descends to the symmetric quotient S2 [EX].

Corollary 11.4 (of Proposition 10.3). Suppose that r > 2. The operations

Sq' : [E,.X] - [ES2X]S'i

have indeterminacy vanishing by [E2 r-2X1]2j (and thus no indeterminacy at E2). They

are linear maps with linear indeterminacy. Now suppose that x E [Er X]t. Sq'x = 0 unless

min{t, r} < i < s + 1, and this vanishing occurs without indeterminacy. In any case,

Sqhx survives to [E2 r-1X 1ti, and the following equation in [E2r-1 ?121 holds (without

indeterminacy):

d2r-1(Sqix) = Sq+r-1(drx).

The notion of top operation has shifted: Sq+ 1 x is the top operation, it equals the

product-square x x x, and in particular, has no indeterminacy. Finally, SqOx = 0, SqIx = 0

when t > 0, and Sq x = 0 when t > 1.

Corollary 11.5 (of Proposition 10.4). At E1 , the operations Sqh : [E1X]s -+ [E1 S 2X]Sj 1

have no indeterminacy, and commute with d1 for each i. They need not be linear. Suppose

that x E [E1 X]'. The top operation Sq'+1 x need not equal the product-square x x x on E1 ,

and Sq'+2x need not vanish, instead equalling x x djx on E1 . For i > s + 2, Sqix = 0.

Finally, Sqix = 0, and Sqx = 0 when t > 0.

Corollary 11.6 (of Proposition 10.8). Fix r > 1. The potentially multi-valued function

Jv : [ErX]s ---+ [ErX]s++1 , defined when 2 < i < max{n, t - (r -

is linear with linear indeterminacy whenever i < t. It is a single-valued operation when

2 t - s and r = t - i + 1;

dr (x) + 6 (drX) = (x 0 drx), if i =t - s, s = 0 and r > 2;

0, otherwise.

If 2 t - s and r = t - i + 1 the set of values

148

of the left hand side coincides with the set of values of the right hand side. Otherwise, this

equation holds modulo the indeterminacy of the left hand side.

The only potentially non-linear operations are

6 : [E1X]X -- + [E1X]'t and 6: [EX]' -+ [EX]ot.

They have no indeterminacy and satisfy 6'(x + y) = 67(x) + 67(y) + bt(x 0 y).

Corollary 11.7 (of Proposition 10.9). For 2 < i < t - s, the operations 57 : [EX.] -- +

[E.X]s7+ 1 agree with the homotopy operations 6i : rt-,X -+ rt-,+iX on the target of the

spectral sequence. Similarly, the product at [EX] agrees with the product on the target.

It seems likely to the author that this is a complete description of the natural operations

on the BKSS in sMom.

Although we do not use the following operations in this thesis (as we do not consider the

BKSS for simplicial Lie algebras in detail), we note that when C = ie or C = fier there

are operations:

P ([EX]t "-L4 [Eq2FCX] i ---+ [ErX ]i+)

Phi: ([ErX]s S!kV [Erq2FeX]~[ s k _ [ErX]sj~1)

[,] : (ErX]s 0 [EX]t: / [Erq2F' X] js| + [ErX1 ,++

with the A' multi-valued functions, defined when 1 < i < max{n, t - (r - 1)}, and single-

valued whenever 1 K i < min{n + 1, t + 1 - 2(r - 1)}, and the Ph' multi-valued functions

with indeterminacy vanishing by E2r-2, and which equal zero unless min{t, r} < j 5 s. We

will not be able to prove a version of Proposition 11.2 in the present work, since we have

not derived a version of 8 for the categories sYie and sYier. Nonetheless, these operations

will satisfy analogues of Corollaries 11.3-11.7.

The purpose of rest of this chapter is to give the necessary constructions to prove Propo-

sition 11.2, so that in the following, we will work only in the category C = Wom. However,

the constructions, including of the following two- and three-cell complexes, generalize to the

categories of Lie algebras.

11.4. A chain-level construction res inducing HC

Let C = Wom. In 6.5, we defined

eHC : B4+ BYHX B-- HX,

149

and in 8.3 we used this map to define Steenrod operations and a product on HJ*H*X.

We will now construct, at the level of Radulescu-Banu's resolution, a map

Xs A L Xs _4 VXs

which, under the isomorphisms of Theorem 4.1 and Proposition 3.8, induces the map He

on cohomology:

H*(XS A L XS) (resO)* H*(VXS)

t tBs H*gX Y BsH*gX : -* Bs H*gX

We will need to abbreviate a little for the sake of compactness. Fix a cosimplicial degree

s. Write X for Xs, V for VXs, V for VXS, dots for categorical products, and superscripts

for categorical self-products. There is a diagram

_( 2 c_(V X) c(V.V) Vc(Eid)o3 c v c(addor2) c(add-T) c(add)

(EijV2.~do2 -2 addT - add -c(X LJX) _X. (X )>V ....... .....> 7 ..........>

in which we define 7 es to be the composite of the horizontal solid arrows. The sub-diagram

consisting of solid and dotted arrows strictly commutes, and we will define *, to be the

unique map up to homotopy such that the full diagram homotopy commutes, after showing

that the composite c(X Li X) -+ V is null. The maps defined here need a little clarification,

during which we will resume writing cosimplicial degrees:

c(c, Id) o : (c(Xs) 2 4 cc(XS) 2 c(idd) c(c(XS) 2 - c( :)i2) cSLd) c((X) 2 c(Xs) 2))

c(add o c2) : (c((X,) 2) c2 c((VXS-1)2) c(add) c(VXS-) = xs)

Fortunately, the fact that the diagram (without the dashed arrow) commutes is obvious: the

small triangle commutes by counitality of 3, and the three squares commute by naturality

of c : c - id.

Proposition 11.8. The map Ges induces the map HC on cohomology, and descends to a

map X : 7L 7 -> V as suggested by the dashed arrow above. This map induces the map

* on homology.

Proof. Under the isomorphisms of Propositions 3.5 and 3.8, if we apply 7r*(DQC(-)) to the

150

solid maps in this diagram, we obtain (abbreviating H5 to H):

(HX)Y 2

(HX) 2 H (HX)Lu(HX)u 2 : (HV)u 2LHX - HVLHV HV(id,id) doudouwLp p+iUdo WS+1

HX x HX

One observes that the horizontal composite is the very definition of HC. We know from

6.5 that He factors through the smash coproduct, which is how we were able to fill in the

dashed arrow on cohomology.

In order to obtain a map *s it is enough to check that the composite c(X U X) -+ V is

null. However, as V is a GEM, a map into V is null if and only if it is zero on cohomology.

We have just stated that the map He factors through (HX)22 , which is to say that the

composite HV - HX x HX is zero. E

This map res is very rich, but it will be important to note that postcomposition with

e destroys much of that richness. That is, reading off the dotted portion of the above

commuting diagram:

Lemma 11.9. The map c o Ces equals the following sum in homsv (c(X 2 ), V):

(F o c(add) o c(c 2 )) + (T o rio e) + (To 7r2 oe),

where the 7ri are the two projections Xx,2 -+ X.

11.5. A three-cell complex with non-trivial bracket

Let C = Wom, and fix t, t' > 1. There is a map Se -+ 5i U Si, sending the fundamental

class zt+t' to the shuffle product of the two fundamental classes in the codomain:

Zt+tl -t (V(Zt 9 z,)),

where At is the structural pairing in Wom. Consider the complex Jt,t, formed as the pushout:

Se t/ Se Li Se

CS" +> J t,t'cS+t, l

The left vertical is evidently almost free (and thus a cofibration), and thus its pushout, the

map S Li S, -+ JI, is almost free. The generating subspace Vt+t'+1 C (Jt,,,)t+t'+1 has a

151

(t+t'+1)-dimensional generator htt, the image of the cone class h in (Sf' )t' + (c.f. @2.5).

Moreover, the object Jt,tl is cofibrant, and htt becomes a cycle in QCJt,t,, since diht,t = 0

for i > 1, and doht,t := zt+t, which we have identified with the decomposable element

pt(V(zt 0 zt,)) in passing to the pushout.

The homology long exact sequence shows that H. Jt,t, is three-dimensional, containing

classes zt, zt, and httI. Moreover, there is a co-operation A on HO dual to the S(Y) structure

map on cohomology, and we prove:

Proposition 11.10. Under A : H2Jtt - (S2H'J,t)*_1, ht - zt 0 zt + zt/ 0 zt. All

other co-operations on H2Jt, are zero.

Proof. The representative g has the property that do(g) = p(V(zt 0 ztl)) and di(g) = 0 for

i > 0. By Lemma 6.3 and the description of que in 3.10:

Oe(g) = que(p(V(zt 0 zt'))) = tr(V(zt 0 zt'))) C (S2 QCJt,t,)t+t.

11.6. A chain level construction of je

Let C = Wom. We can use the cofibration just defined to construct, at the chain level, the

image under

je: PrHe-coalg(HY) 0 PrHC-coalg(HZ) -- + Pr H-4oalg(HY TA HZ)

of a tensor product oz 0 3 of spherical homology classes. Abbreviating HC to H and

PrHC-coalg to Pr:

Proposition 11.11. There is a function

F : home(SC, Y) x home (St,, Z) -- + home(Jt,t, c(Y x Z)),

natural in Y, Z C sC, such that the function

F : home(St, Y) x homse(S ,, Z) -+ 7rt+gt+(Q c(Y x Z)) =: H+ +1 (Y x Z)

defined by F(a, /) := H'(F(a, L))(ht,t') makes the following diagram commute:

hom e(SC Y x hr2> Pr(HY)t 0 Pr(HZ)t, > Pr(HY -A HZ)t+,+homC(SC %Z)

F id+T

Ht+t+1(Y x Z) : (HY X HZ)t+t+1 > (HY A HZ)t++1

(S2 H(Y x Z))t+t, (S2 (HY x H Z))t+t,

152

The south-westerly arrow in this diagram is composite of the tensor product of the maps

Pr(HOY)t C HOY -- H,(Y x Z) and Pr(HJZ)t C HOZ -+ x Z)

followed by (id + T) : (H.(Y x Z))0-*+ S 2 HO(Y x Z).

Proof. The value of 1 on (a, 3) is defined as follows. Construct canonical lifts (c.f. 3.6):

c1(Y x Z) c1(Y x Z)

and (00)

SF Y X Z St Y X Z(aO) (00)

and then form the commuting diagram

S+t' SL ) c1(Y x Z)

CSJ, >J tt - - - - - - -Y x Z

0

The reason that the zero map CSt'-+ Y x Z makes the outer square commute is that the

composite SO t' -+ Y x Z vanishes, as it sends zt+t' to p(V((a, 0) 0 (0,3))) = 0 E Y x Z.

Corresponding to the right square is a map Jt,' -- + c2 (Y x Z), and the composite with

the cofibration c2 (Y x Z) -+ c(Y x Z) is F(a, 3). This function F is evidently natural in

Y and Z, and so then is F.

The required commuting diagram consists of a square, a triangle and a hexagon. The

square commutes as the horizontal arrows are maps in HC-coalg, and we can see that

the triangle commutes because we understand the HC-coalg structure of H,(Jt,t') (and

H'(F(a, 0)) is a map of C-H*-coalgebras). As all of the maps in the hexagon are natural,

we may check that it commutes on the universal example alone:

(zt, zt') E homse(St, St) x homse(S, S,).

That is, it is enough to check that the following hexagon, with a one element set at the top

left entry, commutes:

{(zt, Ztl)} hur®2 Prt(HSO) 0 Prt, (HSO,) > Prt+t'+(HSC K HS,)

F prjinc

Ht+t'+1(SO x st,) L > (HSO x HSt,)t+t'+i :: (HS 7 HS, '

In this diagram, j* and inc are isomorphisms of 1-dimensional vector spaces, so it is enough

153

to check that r(F(zt, zt')) does not lie in the kernel of proj, i.e.:

r(F(zt, zt')) V (HSt Li HSt01)t+t'+1 = Ht+t,'+1S G Ht+t'+1St, = 0

yet A(r(F(zt, zt'))) = zt 0 zt, +zt, 0 zt # 0, using the commuting square and triangle already

established. E

We record here a useful calculation:

Lemma 11.12. For a : Si - X' and :S, -+ X', the composite

S f -+J ,3 c(Xs x XS) c(addo E2)) y

equals 2i Li e0. In particular, (c(add o (e 2 )) o F(a, ))(pV(zt 0 Zt,)) = 77(( 0 e/3)).

Proof. We may calculate the restrictions to the two summands individually, and by symme-

try, we need only consider:

S( > Jt ' c(X9.XS) C(VXS-1-VXS-1) C a c(VXs-) -

The composite Jt,t, -+ c(VXS-.VXs-~) equals F(ca, c/), due to the naturality of F. By

definition of F, the composite S -+ c(Vs- 1 -VXs 1 ) equals (ea, 0). The naturality of the

operation a e 5 finishes the proof. E

11.7. A two-cell complex with non-trivial P' operation

Let C = Wom. In this section, we give a construction of a two-cell complex whose cohomology

has a P' connecting the two cells. Fix t, i with 2 < i < t. There is a map S - S defined

by

zt+i - /('7t-i(zt 0 zt)),

where ft is the structural pairing in C. Consider the complex Et,i formed as the pushout:

St+i - t

CS4~ ESt,i05t+i

By the same observations as made in 11.5, this map is a cofibration, and H2Et,i has

cohomology spanned by zt and ht,i in dimension t + i + 1. For dimension reasons, zt is

primitive. On the other hand:

154

Proposition 11.13. In HCEt,i, P'z* = h*

Proof. We will calculate the action of (Pi)* and A on hti. By the same methods as in the

proof of Proposition 11.10:

=(g) = tr(Vt-i(zt ® zt))) E (S2QeEt,,)t+,

which represents oXtzt. so that the defining equation

(0e)*(ht,j) = Z, rr(1 + T)(yj 0 zj) + Ek J ((Pk)*ht,i)

degenerates to o-i((Pk)*ht,i) = Oxtzt.

11.8. A chain level construction of 0*

Let C = '&om, and recall the linear maps

* :V -+ (CHfom-coalgVlt+i+ defined when 2 < i < t

of Proposition 8.1. After stating Proposition 8.1, we explained that we would define a

non-linear function

0* : Vt -- + (CHom-coalgV) 2t+ 1 defined when 2 < t

using the Proposition 11.14. Thus, in the following proposition, the final statement holds

by definition when i = t.

Proposition 11.14. For 2 < i < t, there is a function

Z7: homv (Kt, W) -- + home(Et,i, cKe W),

natural in W C sV, and satisfying G(a)(zt) = a, such that the function

G : homv (Kt, W) - 7rt+i+(QecKeW) =: Ht+i+i(KeW)

defined by G(a) := H,(G(a))(h) descends to a function

G: irtW -+ Ht+i+1(KeW),

and, whenever 2 _ Se a >0~ ci-(t(Z)t(KW)

CS" E - - -- KW

0

Corresponding to the right square is a map Et,i -i c2(KW), and the composite with the

cofibration c2 (KW) -+ c(KW) is ?(a). This function 0 is evidently natural in W, and so

then is G. In order to check that the resulting function G descends to 7rtW, suppose that

al, a2 E homsv(Kt, W) are homotopic. Choose a homotopy a : A' ® Kt --+ W between a,

and a2. Using the generating cofibrations included in 3.6 and the same technique as used

to define 0(a) produces a homotopy A' 9 Oti -+ cKCW between G(al) and G(a2).

The calculation of G is vacuous when i = t, since G was used to define 0*. when 2 K i < t

is natural in W C sV, so may be checked on the universal example zt E homv(Kt, Kt). As

0(z) is a map of C-H*-coalgebras:

(Pi)*G(zt) = { , if j # i, 2 < j (t + i)/2;

hur(zt), if j = i.

Moreover, AG(zt) vanishes since i < t. These conditions suffice to identify G(zt), as, by

construction, G(zt) lies in quadratic grading 2 of the cofree construction:

G(zt) E q2Ht+i+1(K() = q2 CHC-coaligzt}.

11.9. Proof of Proposition 11.2

Let C = Wom. Proposition 11.2 follows immediately from the following two commutative

diagrams. In each, the bottom row is that used to define the cohomology operations on the

derived functors with which the E2-page can be identified, and the top composite is that

used to define the spectral sequence operations (after applying N* and using the inverse of

the composite of Proposition 11.1).

The commutative diagrams that follow are necessarily large, and at various points

throughout the following two Propositions and their proofs we will use the following ab-

breviations: X for Xs, VX for VX8 , VX for VXs, R X for R iX, H for H*, 7r, for 7r*, Q for

QC and Pr for PrHC-coalg

156

Proposition 11.15. There is a commuting diagram (writing t = t + t'):

7r(I) md ) > 7rv(q2FCIs) 7r >r 1v( ( :co" I+

zig-zagcsn

4 hur®2 Hrf(R Xs) 2o 7r + s(VX8 )2! hur 02

Prt(HXs) 0 Prt,(HIs) H -Prt+1(HIs HI) > Prt+l(HVX8 )

Proposition 11.16. Whenever 2 x 7 Tr~ ( 2 CX S) > 7rV g V"

-=I hur02 0 * CCcolI hur 02

Prt(HX8 ) - > (Ccoalg (Pr(HXs)))t+i+i H TI+ V;- Prt+i+l(HVXs)

Proof of Proposition 11.15. It will help to modify and augment this diagram a little. Indeed,

for each cardinality one subset {(a, /3)} C home(S2, X) x homse(St,, X), there is a diagram:

{(, )} V 7rt(q2FCX) -- * > irt( X) : co"" 7t+I(VX)

eo(hur®2 ) F 7rt+i(QVX) f zig-zagA

- 7r. (Q0)Prt+i (HX T\ HX) Prt+i(HVX)

r (0 (Qe)(HX - H)t+l +1rtQ(cof) > * rt+(QVQ H_+_V_

.7r.

(HX x HX)t+ 7rt+iQc(X x X)

Although all of the arrows in this modified diagram have already been defined, we've dec-

orated some of them for emphasis. It will be enough to check that for each (a, 1), this

modified diagram commutes, since the collection of such (a, 3) will exhaust all of the pure

tensors in irt(X) 0 7rt,(X). What we need to prove is that the large rectangle consisting of

wavy and solid arrows commutes.

The composite of the dotted maps equals the composite of the wavy maps, by results

above. That is, Proposition 11.11 states that the two composites {(a, #)} -+ (HX-AHX)t+i

are equal. The content of Proposition 11.8 is that the small triangle and square at the bottom

of the diagram each commute, and the two composites Prt+i(HX A- HX) -+ Ht+iVX are

equal. Finally, the two composites Prt+i (HVX) -+ 7r+j (VX) are equal, by Lemma 3.6.

Thus the image of (a, 1) under either the wavy or the dotted composite equals the image

157

of htt c 7rt+1(QJt,t') under the composite

QJt,t' Q4 Qc(X x X) QLeT QV 2 QVX =VX,

which, by Lemma 11.9, decomposes as the sum of the three maps Uo c(add) o c(c2 ) oF(a, /)

and T o 7ri o c o F(a, ) for i = 1 and 2. The composite i 0 E oF(a,/) : J, -- + X, by

construction of F, is the (dashed) map out of the pushout in the diagram:

Ste: Se U StttV(zt &zt1) UO

CSC t

0

Now htt is in the image of the map CSC -+ Jt,t, and so maps to zero under the dashed

map to X. Similarly, the composite r2 o E 0 F(a, 3) vanishes on httI. Thus, the image of

(a, 0) under the dotted composite is represented by

A := (U o c(add o E2) o F(a, 0))(ht,t').

Consider the following commuting diagram:

QF(a,#3) Qc(addoE 2 ) Qht,t, E QJtt' - 1, Qc(X x X) > > QVX E A

dol dol doF((,#) c(addoE

2

AV(zt 0 Zt') E Jt i > c(X x X) > x T p(V( a 0 E0))

The element ht,t, E N'+1 (Jt,t') may been used to populate the whole diagram as shown. To

understand the images of htt at either end of the bottom row, note that doht,t = [V(zt&zt')

by construction, and Lemma 11.12 states that under the maps of bottom row, pV(zt ® ztl)

maps to p(V( 0 e#3)).

The data in the bottom right corner of this diagram demonstrates that OcnAl E 7r(R IX)

is represented by T(V( 0 c#)), which suffices, as a ~ 6Q and c ~ . F

Proof of Proposition 11.16. Choose a representative a E home(SC, X). Then, setting W =

QX~ 1 in Proposition 11.14, we obtain a map G(Ea) : Et,i -+ X such that (0 o hur)(a) is

represented by

(T o G(Ea))(ht,j) E Ntviisl(VX).

158

We populate the following commuting diagram using the element htj E N +i+1(Et,i):

ht,j C N v+i+ - U > Nv+i+1 (VX)D (77 o G(Ea))(ht,j)do do

A7t0i(zt 9 zt) E ZN'+iEt,i > ZNv+i 3X pV ti(3 ® ia)

Here, the value of doht,i is known by definition of Ot,j, and the fact that G(ca)(zt) = 23 allows

us to calculate (G(ca) o do)(ht,j). Finally, in order to calculate 8conn(67 o hur)(a), we find

a preimage under N+i+ 1 X - - Nv+i+1VX of the representative (rj o G(ca))(ht,j), and then

apply the differential do. We may use the preimage C(Ea), which maps to pVti( a 0 ) E

Nt+iRIX under do. This is homotopic to pVt-j(a 0 a), which represents TTJpct (a). l

159

160

Chapter 12

Composite functor spectral sequences

12.1. The factorization of QW(n) and resulting CFSSs

It will be important for us to identify the derived functors HQ()X := D(L*Qw(O)X) for

X E W(O), in order to determine the E2-page of the BKSS for a connected simplicial

commutative algebra. More generally, we will now present a spectral sequence whose goal

is to calculate H (fl)X for X E W(n). This will be a CFSS analogous to Miller's spectral

sequence in [43, 2]. We note in 12.6 that our CFSS is also a reverse Adams spectral

sequence, and discuss Goerss' use of such a spectral sequence.

The factorization of QW(n) we will use is of course

S(n) QL()w(W) =(n) -- + Z(n) :-- Vn

There is an added challenge in this context - indeed, the available factorization of QW(n)

is through a non-abelian category. Thus, the standard technology for CFSSs does not

apply, and we must use Blanc and Stover's methods [3]. They observe that the left derived

functors L.QU(n)X are calculated as the homotopy groups of a simplicial object in L(n),

namely QU(n)BW(n)X, and as such, they have the structure of a r(n)-II-algebra. That is,

they form an object of W(nr+ 1). After verifying that the functor QU(n) satisfies the requisite

acyclicity condition (indeed it preserves free objects), one may apply [3, Theorem 4.4]: there

is a spectral sequence, with Er E V+2'

E 2(HW(n+l))I~un))~ 2 ~=~.~[E X]'+,.,s = ((H, " (L*QU(n))X)1a+,.,i- ((H, 4')X)'n2s+,a..s

If UW,U: W -+ U is the forgetful functor, resulting from the fact that an object of W(n) is

in particular an object of U(n):

Proposition 12.1. For X E sW(n), the groups L*Qu(n)X are isomorphic to H %/U

161

the U(n)-homology of the object of sU(n) underlying X.

Proof. We may take X to be almost free in sW(n), and calculate L,Qu(n)X simply as

7r*QU(n)X. Then X, viewed as an object of sU(n), is levelwise free, but potentially not

almost free. We need to show then that 7r*Qu(")X does indeed calculate H X whenever

X E sU(n) is levelwise free, which is to say that the map QU(n)BU(n)X - QU(n)X is a

weak equivalence in sV. For this, Qu(n)BI(n)X is the diagonal of the bisimplicial vector

space QU(n)Bq)Xp, and we use the spectral sequence arising from filtering by p. As X is

levelwise free, the E1 -page is isomorphic to the chain complex Np(Qu(n)X), concentrated in

q = 0.

We will prefer to work with the dual spectral sequence, which has Er C Vn+2

[EWX],n2 = ((H.(fl+1))(HY(n))X)sn+2.,Si == X] sn+2, --,S1

These two spectral sequences are respectively the homotopy and cohomotopy spectral se-

quences of a certain object of ssV+, with which we will need to work directly. Indeed, in

12.2, we will define a comonad 9 on sC(n), and, for X E W(n), we will use the object

Qn)BO'L (E ssV+ where L:= Qu(n)Bw(n)X E sLC(n).

The identification of El follows from Lemma 3.1 and Propositions 12.1 and 12.2.

Before we do, we will recall Blanc and Stover's constructions, and imbue them with

certain extra structure that will be reflected in the spectral sequence.

12.2. The Blanc-Stover comonad in categories monadic over

F2-vector spaces

Fix an algebraic category C, monadic over a category of graded F2 -vector spaces V. As

we are working over a category of vector spaces, rather than a category of graded sets, we

can find further structure on the following comonad on sC defined by Blanc and Stover.

While they use the notation 'W' in [3] and '"' in [55], we will use the symbol '0' to avoid

notational confusion. In our context, Blanc-Stover's comonad , applied to L C sC, is the

pushout

H SEsph(C) Syon USesph(C) Sxy:CS-+L x:S-+L

HSesph(e) CSy > 9Ly:CS-+L

162

The subscripts are just notation to distinguish multiple copies of S and CS for each sphere

S E sph(C). The top horizontal map sends the sphere Sy,, isomorphically onto itself. The

left vertical map is the coproduct of copies of the inclusion in : S - CS. The effect of

taking this pushout is to modify the coproduct S. of spheres by attaching the cone on S"

once for each nullhomotopy of x E L.

It will be useful to write hy for the image in NWL of h E N.CSy, and similarly, zx for

image in ZN9,WL of z E ZNSx. Indeed, recalling the discussion in 3.1, the data of S E

sph(C) with a map S -+ L is equivalent to the data of a homogeneous normalized cycle of L,

and similarly, S c sph(C) with a map CS -+ L is equivalent to a homogeneous normalized

chain of L which is not in dimension zero. From this viewpoint, if we write hg(ZNL) for

the homogeneous normalized cycles and hg(N>1L) for the homogeneous normalized chains

of L not in dimension zero, the pushout may be written as

H yEhg(N> 1 L) Sdy H XEhg(ZN.L) Sx

Hychg(N>1L) 0 SY > L

We will now show that 9L is homotopy equivalent to a coproduct of spheres. Indeed,

let

hg(BNL) = im(d : hg(N> 1 L) -* hg(ZN*L)),

and choose a section f of the surjection d: hg(N> 1 L) -- hg(BN.L). Then 9L contains a

contractible subobject, the pushout

HxEhg(BN.L) Sx HxEhg(BNL) Sx

HxEhg(BN.L) CSf(X) > C

whose inclusion is a cofibration. Then

WLICo [J CSY/S Li H s.yEhg(N;>L)\im(f) (xEhg(ZNL)\hg(BNL)

where we have written 'A/B' for the pushout of a cofibration B -+ A along the map

B -+ 0, using the cofibrations Co -- +1 9L and in : S -- + CSy. As CS/S is isomorphic to

the sphere of one dimension higher than S (consider the construction of 2.5), this shows

that 9L is homotopic to a coproduct of spheres.

The promised comonad structure maps e : 5L -+ L and A : 9L - 2L are deter-

163

mined by:

e(hz) = x, E(zy) = y, A(hx) = hh,, and A(zy) = z. for x C N,+L and y C ZNnL.

We would like to find a subspace of gr,(WL) which freely generates it as a C-H-algebra. Even

better, we have the following rendition of an observation used in [3, Proof of Theorem 4.2].

We give the proof since we will need to be explicit about some parts of it in what follows.

Proposition 12.2. For L C sC, 7r,(BW'L) is an almost free (monadic over V) simplicial

C-H-algebra weakly equivalent to 7rL.

This differs from the observation implicit in [3, Proof of Theorem 4.2], in that we show that

all the structure maps of 7r, (BWL) c s7rC except for do preserve vector spaces of generators,

rather than sets of generators.

Proof. That the augmentation to rL is a weak equivalence follows from Stover's result [55,

2.7]. The only change from Blanc-Stover is that 7r,(BWL) is almost free over the category

V, rather than the category of pointed sets.

During this proof, for any set A we will write F2 {A} for the vector space generated by

the symbols a for a E A. Suppose that M G sZ(n). There is a natural map

d, : F2 {hg(N>1M)} -- F2 {hg(ZNM)},

and a natural monomorphism

a : ker (d.) -+ r,(M)

defined by

a(xi - xo) = h,, - hxO, for X1, x2 E N>1M with dxi = dx 2.

Moreover, there is a natural map 0 : lF2 {hg(ZNM)} -- + ir,(WM) (which is not monomor-

phic) defined by

()= , for x E hg(ZN.M).

From the above expression for 9M/Co, one sees that im(oz) and im(3) are linearly inde-

pendent subspaces of ir.(9M), and that r,( M) is free on im(a) e im(3). Moreover, if

M -+ M' is a map in sC (n), then the generating subspaces are preserved by the induced

map rw9M -- + 7rVM'.

Applying this analysis to 7r.BE'L E s7rC, every face and degeneracy map except for

so and do preserves the generators. In order to check that so preserves generators, we

must see that the comonad diagonal of 9 sends the subspaces im(aL) and im(OL) into

164

the subspaces im(aWL) and im(qL). That im(OL) maps into im(OWL) is immediate. For

im(aL), the image of h., - hx0 under the diagonal is hh. 1 - hhO, which is in im(aWL), since

dhx1 = Zdxl = Zdxo = dhxO. II

12.3. A chain-level diagonal on the 9 construction

We have seen, for M E sC, that 7r,(WM) is a free object in irC. As such, there is a diagonal

pre : 7r*(OM) -- + 7r('M) U 7r*(WM).

In this section, we will describe how poe is the map on homotopy induced by a morphism

ov : 5M -- > M U WM in sC, and construct a map w related to the map ,e of 6.5.

In order to construct a map pw, each S E sph(C) equals S - FCK for some K as in 2.5

(with indices omitted), and we construct a commuting diagram:

S -O S U Szf (P2 jinU'Z

CS S CSuCSby applying FC to

K Ka >KK

C m C 'UznCK-> CK (DCK

The maps p and 02 can then be applied respectively to all of the sphere and cone classes

appearing in 9M. To understand the effect of pg on homotopy, it is enough to identify

where the generators of 7r, (M) are sent to in r. (9M) L 7r, (9M), which is easy. The theory

of this map mimics that presented in 6.5, as intended, and we list some of its properties

here, with proofs omitted.

Lemma 12.3. 9M is naturally a (strict) commutative cogroup object, having comultiplica-

tion map p, counit map 0 : 9M -- 0, and inverse map id : WM -+ 9M. In particular,

hom( M, -) takes values in F2 -vector spaces.

Writing E for the group operation on home(O'M, M'), we have the following:

Lemma 12.4. For maps f, g : 9M - M' we have

QC(f E g) = (QCf + QCg) : QC(WM) - QCM'.

Proof. It is enough to check that QC() -

map QC( 9M) -+ QC(WM) e Qe(09M).

the construction of WM to direct sums of

both precisely the diagonal map.

Qe(WM) - QC(OM U M) equals the diagonal

For this, QC converts all the colimits involved in

simplicial vector spaces, and QCcp1 and QC,02 are

165

Now let q denote the following composite:

0?: 2M 22!> (072M)u2 "-+(lb u

where a, b : -2M _- (0,M)12 are the composites

a : 012M ff : (q2M)U22 (gM)U2

b : 0?2M >(WM) **>(OM)U2

Thanks to Lemma 12.5, ( factors through the smash coproduct, defining a natural map

: W2M - (WM)V 2.

Lemma 12.5. The composite g 2 M (WM)U2 -- + (M) x2 is zero.

Proof. This follows from the observation that both composites (id Li 0)( and (0 LI id)Zq

equal e : 02M -+ 0M. El

The desired property for q is then the following lemma (involving the natural isomor-

phism i of Proposition 3.2, and the almost free structure given in Proposition 12.2).

Lemma 12.6. For L E sC, we have (i o e) = wr( W), i.e. a commuting diagram:

7r* (BW L) 7r* > r*((B' _L) V2)

(7r*(BO'_ L))Y2

Proof. In view of the short exact sequences of Proposition 3.2, this is equivalent to (iole) =

7r*(W) :7r(+L - r*((0L)U 2 ), which holds as (i o oe) =r r( p). El

Lemma 12.7. (di)Y 2(g = $ fdj+1 for i > 1, and (do)Y 2 (O = (eqdo) E3 ($Wdi), so that the

map QC$, induces a degree (-1, 0) bicomplex map:

N*N*(QCBT L)Sn+2,Sn+1 -+- N*N*(Qc((B: L)Y2) )S2-1,81

As in 6.5, we will use the composite double complex map

Oq:= jZ(n) OQZ(n) q : NghNJ(QC(n)BW L)ts2,..l -+ NvN(S2 (QZ(n) BW L)) t+-1s+,..s2 f.l.ow.. : NnN2Q'(sB~L ..).

in what follows.

166

12.4. Quadratic gradings in the CFSSs

We will say that an object X E C, where e is any of W(n), U(n) or C(n), is quadratically

graded if the underlying vector space of X is equipped with a quadratic grading such that

the action map F2X -+ X preserves quadratic gradings (i.e. is a map in qV+). Recall that

Fe is in fact a monad on qV+, by Lemmas 6.10 and 7.2. There are evident categories of

quadratically graded objects in these three categories, which we write as qW(n), qu(n) or

qC (n), and the various homology and cohomology functors can be enriched to functors

He : s(qC) -+ qV++ 1 and He : s(qC) -- + qV+ 1 .

Similarly, the categories Mv(n + 1) and Mh(n + 1), in which H,(Q) takes values, can both

be enriched in this way, and if X E qW(n) then HQ(n X is an object of qMv(n + 1) and

qMh(n + 1), and H X is an object of qW(n + 1).

Thus, if X E qW(n) then the CFSS

[ESX]Sn+2,.,Si = ((H (n+1 ))(Hy(n))X)Sn+2,...,Si = [EOHnX]+ 2 .---,S

has both E2 and target quadratically graded. Because all of the cohomology and homotopy

operations constructed in 5-8 are formed at the chain level using quadratic operations, it

is not hard to check:

Proposition 12.8. If X E qW(n) then the CFSS is quadratically graded:

q [EX] sn+2,...,Si =qk(HI=+))(HU(n))X)Sn+2,...,s => qk[EoHl(f)X] Sn+2,. ,Si

12.5. The edge homomorphism and edge composite

For X C sW(n), the spectral sequence

[EWX]Sn+2,..., = (( )X)Sn+2,...,Sl => [EOHJ XJSn+2,...,)8

has edge homomorphism

H )sn+1,.,- [EoH= X?'.si ~ [E X]OSn+1,...S C [EOX]0'sn+1,...,Si

which we may compose with the inclusion

[EX]'...'.'i = (D(Qw(n+1)Hf (n)X))Sn+1,...,Si C (H.iX)Sn+1,--.,Si

167

to form the edge composite:

(H%,X,"1-' -+ (H* X)tn"+1' 7

Proposition 12.9. Suppose that n > 1. Then the edge composite commutes with the vertical

Steenrod operations of Proposition 8.6:

S sn+1,---,s1)81 * sn+1+1,sn+i- 1,2s,- 1,...,2si1S(H nX) s (H* X) 2sl 1.

edge comp. edge comp.

S n .is i S q ?, + 5 5 2(HQX)32+,..,3 s">(H* X)Sn+ +1,sn+i-1,2s,,-1,I...,12si

Setting n = 0, suppose that 2 < i < t. The same composite commutes with the 6v-operations

of Propositions 8.2 and 8.3:

6V

A HoX)s (H, ( X)s"|+1

edge comp. edge comp.

( H OXs ( H O X )8"+1

Proof. For this proof, we will suppress the '(n)' notation, as the proof is the same for all

n > 0. We will also suppress all internal gradings, and write * for the grading sn+1. The

edge composite is dual to

HX := 7r*(QWIBWX|) d ro7r(Q B'Q|B X\) + 7r*(QU|BvXI) - HUX.

Abbreviating further by setting D := QuIBWXI and C := QEBQUjBWX, the map z-

sends the class of x C ZN*D to y- E 7rh7rVC. This assignment does not produce a well

defined map 7r*D -- + Nh 7rvC, as if y C ZN*D represents the same class as x, Z7y need

not equal i-T in N h7r[C: we only know that zx-y = 0 C N h7,rC. Fortunately, the element

zz--z, - zz E N 7rvC provides a homotopy between z - and zx - z in N h7r[C:

o (ZZ2-2 - zzx_,) = zx - ZY - zx-y, and d, (zz,-zy - zzx_,) = zX-Y - ZX-y 0,

so that the map z_ is well defined.

We may model the final isomorphism as follows. Write Uw : W -- +' U for the forgetful

functor. For any V C V+, there is a natural inclusion FUV -+ UjwFWV in the category U,

adjoint to the inclusion V - FPWV. This morphism yields an inclusion of bar constructions,

a weak equivalence |B'UjXI -+ U}flBWXI in sU. Suppressing the forgetful functors, for

X C W, we have a weak equivalence QUIBUXI -+ QUIBWXI inducing the isomorphism.

168

Our conclusion is then that the entire composite HUX -+ HWX is the map on homotopy

induced by the composite

QUIBUXI -- + QUIBWXI QwIBwXI,

and the operations we are considering are easily understood in relation to this map. E

12.6. An equivalent reverse Adams spectral sequence

We will show in this section that each CFSS given in 12.1 coincides with a certain instance

of Miller's reverse Adams spectral sequence (c.f. 3.4) for the category s1(n). A reverse

Adams spectral sequence (that for sWom) also appears in [33, Chapter V], and while in both

cases the goal is the calculation of derived functors, the role played by the reverse Adams

spectral sequence in [331 is very different to the role played by CFSSs in the present work.

One might wonder whether the spectral sequence operations defined by Goerss in [331 are

relevant in this setting, and if so, how they relate to the operations we will derive in 13.

The author has not investigated this question.

In any case, we prefer to use the Blanc-Stover resolution for two reasons. Firstly, this

resolution more closely reflects our intention in constructing the spectral sequence in ques-

tion, and secondly, the techniques we use here may be generalizable to other contexts in

which the Blanc-Stover resolution is used to derive a CFSS.

Proposition 12.10. The CFSS applied to X E sW(n) coincides with the reverse Adams

spectral sequence applied to L := QU(n)BW(n)X E s(n).

Before proving this fact, we should remove any confusion about the convergence targets of

these spectral sequences. Indeed, the reverse Adams spectral sequence has target

7r*DQJC(f)B'C(f)L 7r*DQZ4)L = 7r*DQW(n)BW(n)X =: H() X,

where the isomorphism follows from the same acyclicity condition needed to define the CFSS.

Thus the targets coincide, as hoped.

Proof. We will use the Dwyer-Kan-Stover E2 model structure on the category sse, which

originated in [27] for bisimplicial sets, and is reinterpreted for objects of ssC in 13, 4.1.11.

Viewing L as a constant object in ssC(n), each of B(n)L and BL admits an E 2-weak

equivalence to L. Moreover, each is cofibrant. Indeed, BL is cofibrant by construction,

while we must check that B (n)L is M-free, in the sense of [3, 4.1.11.

For this, we use Lemma 2.3. That is, for each q, the horizontal simplicial object

(Bf(n)L),, B(n)L has an obvious structure of almost free simplicial (in p) object,

169

and the generating subspaces are preserved by the vertical simplicial maps. Thus, Lemma

2.3 yields decompositions

V= im(V,- i v, e ... E im(Vpi V,) e (V n Nh B,(n)Lq )

To show that V is M-free, we need to decompose each V into a coproduct of objects

K+ Si E sV,+ up to homotopy, and ensure that the degeneracies are induced up to

homotopy by sphere inclusions. The decompositions of V just provided make this a simple

task. Suppose that V_ 1 already has chosen decomposition as a sum of objects K

up to homotopy. Then if we choose such a decomposition of V n NfBl - )Lq, and use the

p inclusions si : Vp- 1 - V to induce decompositions of the other summands of V using

the decomposition of VI, we have the decomposition up to homotopy that we need.

Now, by factoring the map 0 -+ L by a cofibration followed by an acyclic fibration

B -+ L in the E2 model structure, we can form the solid maps in a diagram in which each

object 'B' is cofibrant:

B ...... B.. L.Aq

BE L L

By the lifting axiom (of cofibrations against acyclic fibrations) we can find the dotted maps,

weak equivalences making the diagram commute. The theory presented in [27] then explains

that the three resulting spectral sequences coincide. The spectral sequence arising from

Bp4()Lq is the reverse Adams spectral sequence of L in sZ(n), and that arising from BEL

is the CFSS of X E sW(n).

170

Chapter 13

Operations in composite functor

spectral sequences

Singer [53] developed a useful theory of products and Steenrod operations in the first quad-

rant cohomology spectral sequence arising from a bisimplicial cocommutative coalgebra.

Goerss used this theory in [33, 14] in his calculation of the category H'om. In the appli-

cations we have in mind, the bisimplicial object

QJ(n)B&QU(n)BW(n)X

will not be a coalgebra. Instead the situation will resemble more the situation of 6.5, where

there was a linear mapoc : QeXs -+ S2(QeXS-1) for any almost free object X E sC, but

certainly not a coalgebra map.

The lack of an underlying coalgebra structure will not stop us from applying Singer's

techniques after we make the appropriate modifications. The idea is to externalize Singer's

operations, so that for every bisimplicial vector space V, there are various external operations

of type:

[ErV] -+ [ErS2 V] (r' > r) and S2 [ErV] -+ [ErS 2 VI

(which we will discuss shortly) compatible at E,, with external operations of type:

H*(D(TV)) H*(D(TS2V)) and S2H*(D(TV)) -+ H*(D(TS2V)).

When V is in fact a bisimplicial cocommutative coalgebra, one recovers Singer's theory by

composing with the map of spectral sequences induced by the coproduct:

[ErS2 V] _+ [ErV].

171

In 10.1 we discussed spectral sequences with indeterminacy, and multi-valued functions.

They reappear in Singer's theory, as some of the operations are constructed as (actual) linear

functions [ErV] -- [Er'S2 V] between different spectral sequence pages. Such an operation

is equivalent to an external operation [ErV] __ [ErS2 V] with indeterminacy r' which also

satisfies a survival property.

13.1. External spectral sequence operations of Singer

We now summarize some key aspects of Singer's work in [53], in particular Theorems 2.15,

2.16, 2.17 and 2.22, and Proposition 2.21. Fix V E ssV with a (horizontal) augmentation

do : V -- + V- 1 . The key construction is that of chain level operations:

Sk : D(TV) -* D(TS 2V)

inducing external operations as in the bottom row of the following diagrams:

7rm(D(_) s'p' x k(D(2-) S27r*(D(V_1)) t"xtM 7*(D(y_)

Hm (D(TV)) .q3,- Hm+k(D(TS 2 V)) S2H*(D(TV)) t" % H*(D(TS2V))

The top rows are the operations arising from the singly (vertically) simplicial object V- 1 ,

as in 6.2. Singer studies the effect of Sk on filtration in detail, determining that it induces

the following operations. For all p, q > 0 and all r > 2, there are well-defined vector space

homomorphisms:

Sqk : [EV]P'q -+ [ES 2 V]p,q+k, if 0 < k < q;

Sq : [EV|P -+ [ E+k-2 S2yp+k-q,2q, if q < k < q + r - 2;

Sq : [E, V]p-+ [E2 r-2g2V]p+k-q,2q, if q + r - 2 < k;

which commute with the differentials (in the appropriate, somewhat complicated sense, c.f.

[53, Theorem 2.17]), and an external (not 'exterior') commutative product operation which

satisfies the Leibniz rule:

Pext : [ErV ]P1,'l 0 [ErV]P2

,2

-- + [ErS2V]P1+P2,q1+2.

Note that the second and third operations are from Er -+ Er1, sometimes with r' > r,

which is to say that these operations have indeterminacy vanishing by Er,, and the implied

survival property.

172

Those operations with domain [E2V] have no indeterminacy, and we reindex them as

follows:

Sqk",.t = Sqk :[E2 V]P' -- [E2S 2Vp,q+k, if 0 < k < q,

Sqex = 0, if k > q,

Sqk = Sq+k: E2 V]P'l -+ [E 2 S2V]p+k,2q, if 0 < k.

Under the identification [E2 V]P 7(DV), the operations Sq,ext are obtained by ap-

plying rP to the linear maps of 6.2:

7r(DV) si +k (S2 DV) - (D

On the other hand, the operation Sq kext equals the composite:

___~ k V2q7p kh GJlext p+k 2 S2 Vhkh pwPk7r P7r qDV 7 4+(S27r*DV2 hn xr +k7r D -- 7p[+k r2qD y

and the pairing Pext : S2([E 2V]) -- + [E2 S2 V] equals:

-7r r((r,*S * wwD V '4 irj(S27rvDV) 7(7r*SD -- +* r*DSV

These operations on E 2 determine the operations at each Er, r > 2. The operations

Sqek commute with differentials as appropriate. Finally, the Sqext stabilize to well defined

maps on Ex, and there is a commuting diagram

[EoV]P'q sqext > [ExS2 V]p,q+k

Sk[EOH*(D(TV))]' sq> [EOH*(D(TS2V))]p,q+k

whenever 0 < k < q, and a commuting diagram

Sq kEVV]pTh sqt > [ EoS2y p+k-q,2q

I l~ Sqkt 4',[EoH*(D(TV))]P'; s. [EoH*(D(TS2 V))]p+k-,29

whenever k > q. These diagrams also serve to summarize Singer's computation of how the

Sqekt interact with the filtration on cohomology.

173

13.2. Application to composite functor spectral sequences

In order to use Singer's constructions in the present work, we will use the map of double

complexes:

O (n) 0 Qf(,) g : Np+,Nq(QL n)B! L)t NpNq (S 2 (s,)B.,SL

to define a spectral sequence map

[E 2 S2(Q L(n)B L L)]p'4'' '.-'j -. . [E2.X].+.1q.,,...,i S

We then define the following internal spectral sequence operations, in each case by the

formula Sqk := o Sqk-

Sqk : [E , X]p'q'S"''"1 -+ [E7X]+1 q+k--1,2s,...,2sj (0 < k - 1 < q),

Sqk : [EpXq'S'".-.'1 - [Ek-q-1 X]p+k-q,2q,2 ,...,2si (q < k - 1 < q + r - 2),

Sq - EX 'p'q''''',1 -- + [E2 -X]p+q,2q,2sn,...,2sl (q + r - 2 < k - 1).

which at E2 we may write (dropping internal degrees) as:

SqQt = o Sq-J: E --E + EP+,q+k-l

Sq = o Sqk- = V/ o Sq q+k-1 : E E+k,2q

if 0 < k - 1 < q,

if 0 < k - 1 <p.

Similarly, we define a pairing:

[ = V5 0 Pext : [EWX]'Sn.'S .. .1 I n.S1+S'

The reader might now guess the key results:

Theorem 13.1. At E2 H(+)H )X, the operations Sqk and p defined here are equal

to the Mh(n + 2)-operations of the same name defined on W(n + 1)-cohomology in 8.3.

Theorem 13.2. At E2 H,(n+1) H* n)X, the operations Sqk defined here are equal to the

M,(n + 2)-operations of the same name defined on W(n + 1)-cohomology in 8.2.

Theorem 13.3. At E,, , [EOH (f)X], the operations Sqk are compatible with the Mh(n+

1)-operations of the same name defined on W(n)-cohomology in 8.3.

174

13.3. Proofs of Theorems 13.1-13.3

Proof of Theorem 13. 1. This proof relies on a commuting diagram, in which we employ the

notation L = QU(n)BW(n)X E s12(n), and abbreviate using 4 = r(n) and W = W(n + 1).

(NhrlvQZBqL)®2 N*(y) (NhQ W7r BqL)®2

(DP -k+l)* (Dp -k+)*Nh ~ ~ ~ ~ N h_ 2 _0)hq)(2

NP+k-((7rvQLBqL)2 ) Nh+k 1 ((QWrBv L) 2 )

N h(Dv)*

NPh+k 7r((Q' BqL)02 ) Ow

Nh+7rvQBL N ) I wNh B

All of the horizontal maps are the isomorphisms of Lemma 3.1. By [53, Theorem 2.23]

(summarized in 13.1), the left hand vertical composite is that used to define the horizontal

operations Sqh on E2 . On the other hand, the right vertical was used in 8.3 to define the

Mh(n + 2)-operations on the W(n + 1)-cohomology groups with which the E2-page can be

identified. Thus, if the diagram commutes, we are done. If we replace the maps (Dj)* in

the top square with (D')*, the same proof applies for p.

What remains is to prove that the bottom square commutes. It may be expanded into

the eight maps in the outer square of the following larger commuting diagram:

7r( (QJ'P+k L) 02) ((D* rvQlCp+kL)02 (Qw vc'p+kL)®2

7r') QWZ )) t jW7rvQi((q'p+kL)Y2) . QW 7v (((9p+k L) V

2) i WMQw((vC'p+k L) Y 2 )

7rvQZ Wp+k+1L :QWgeg~p+k+1L -EQwrvp+k+1L

The bottom left square commutes by naturality of 'y, while the bottom right square is

an instance of Lemma 12.6. What remains is to check that the hexagon commutes. For

notational convenience, write A = p+kL E s1, br : A®2 -+ AY 2 for the Z-bracket, and

br : (7rvA)02 -+ (7rvA)Y 2 for the W-bracket on homotopy.

The source in the hexagon is then QW((7rvA)Y 2 ), the smash product being the coproduct

in W = 7rZ of two copies of 7rvA. Any element of QW((7rvA)Y 2 ) can be represented by a

sum Ek br(- 0 y) + E, with the Xk (resp. Yk) representatives of elements Yk in the first

(resp. second) copy of 7rvA and, E a sum of at least three-fold brackets of elements in the

two copies. This extra term E is annihilated by both jw and 7rv(j) o y o QW(i), so can be

175

ignored. One calculates:

(y" Mojw)(Ekbr(T&k) +E)= E k0 Yk

On the other hand, the map QW(i) is induced by the Eilenberg-Mac Lane map shuffle map

V as in Proposition 5.2, and

Zk br(k 0 ;k) -* Zk br(Vv(xk 0 Yk)) " k(Vv(xk 0 Yk)) (D Xk*Y

The last mapping follows from the fact that (Do)* o Vv = id, as {Dk} is special. l

Proof of Theorem 13.2. We again employ the notation L = QU(n)Bw(n)X E sI(n), and

abbreviate using Z = Z (n) and W = W(n+ 1). Further, write B for the object Bwrv L E sW.

Write Vm for the subspace (Fw)m C B of generators, and V := Vmn NmB. For each m > 0,

write FmB for the m-skeleton of B (c.f. 2.6), which is almost free on subspaces FmVm C Vm.

We must identify the operations Sqg = @b o Sqij_1 with the W-cohomology operations

Sq'v defined in 8.2 using the maps 0'. However, the 0' are defined on the bar construction,

while 0* is defined on the Blanc-Stover resolution. In order to make the comparison, we

will need to choose a sufficiently explicit weak equivalence of resolutions of 7rTL in sW

x : B -- 7*r(BWL).

In order to define x, we recursively define its restriction to the skeleta FmB. Lemma 2.3

implies that in order to extend a (horizontal simplicial) map Xm-1 : Fm-1B 4 7rv(BfL)

to a map Xm : FmB -+ 7rv(BOL), we need only to specify the values of Xm on V4. That

is, we only need to choose a lift in the diagram

VIM - -" - 3- Nm * (BW L)

ZNh 1 B a--- ZN_1 7rV(BWL)

However, in order to actually carry out this process, we will need to record some chain level

information, and we will construct maps into NvB OL, rather than just 7rvBE'L.

It is best to view the domain and codomain of the proposed map X as augmented

(horizontal) simplicial objects, and start by defining X-1 to be the identity of 7rvL. Then

for m > 0, we will recursively construct functions -m : Vm -4 ZNlvBWL, with the property

that im(Tm) is contained in the span of the classes z. for w E ZNB 1_L, so that there is

176

a commuting diagram:

VIM > ZNBiLXM~

In order to do this, one may choose a basis of Vm, and then for each basis element v E VM,

choose a W-expression e for dv, so that

d v = e(shwj) E ZNhIB is a W-expression in various sh.wj E Vm-n,

with wj E V for integers nj < m - 1 and degeneracy operators sQ, : Vj - Vm-i. Then,

from the cycles sa, nj(wj) C ZN2'BfL, form a cycle

erep(s hjT,(wj)) C ZNv(B_ 1 L),

using the explicit formulae of [20, 8] (which is a normalized cycle, as these formulae preserve

the normalized subcomplex), so that

(s jn, (Wi)) = e (s jTn(w)) E 7rv(Bm_ 1L)

= e (s Xn (wj))

= Xm-1(d V) z ZN _17rv (B L).

Our definition of Tm(v) is

Tm(V) : Zere(shx (wj)) E ZNT B"mL.

To check that the class of Xm(v) in 7rv(B L) is in fact in Nirv (BE'L), for 1 < i < m (c.f.

[55, Lemma 2.7]):

dm(V) Zdh_ereP(s, (,j)), and d 1 ereP(s nj (w)) 1Xm-I(d~V) =0.

By construction of the comonad 01, d Tm(v) must itself be null. Thus Ym does induce a

map Xm : Vn -- + Nmi7r*(B'L), completing the construction of X.

Recall that the operations of 8.2 are the maps induced on cohomology by the degree -1

endomorphism 6 of the chain complex NP(QWBW7rvL):

0' : Np1 (QwBw'Tv L)Z9,s,., -+ N2g ,(Qw BwirvL)tIf qwt = Bi- 2smy.t2sh s qsen , rg s

If we write V - Q'C B~fL for the double complex yielding the spectral sequence, -our goal is

177

to identify these operations with the spectral sequence operations

* Sq i-1 : ([E 2V]', S [ 2 s2 v]p,q+i-1 4 [E2 V]p+1,q+i-1

using the equivalence QWy in sV induced by x and the isomorphism -y:

QWX: (QwBwrvL q4 Qw7rv(Bg) -24 7rv(V)).

The composite @g a Sqi-it has been identified as the dual of the composite in the bottom

row of

Njh1 (QWBWirvL)q+iI > Nyh(QWBWir L)q

WX (Sq'IQWX

Nh4(r V4q~- > Nh(7r VS2 V)q+i- 1 ( sqe) Nh( Vq

so that it is enough to prove that this diagram commutes for 1 < i < q.

Given the equations in 6.3 defining the operations (Sq%-1t)*, it will suffice to show that

the composite

(QWB,7rvL)q+i-1 (7rvQ4B +1L)q+i-1 (7iq BLqi-

equals the sum of the composite

(QWB,71rL)q+i-1 -W (2 QwB,*,rvL)q+i-lS2 (QWX) 2 v V 2QC~

S (S r*Q BL)q+i-1 (7r*QBf L)qli-1

with those composites

(QWBV,7irvL)q+i-1 (QWB7rL)q (7rvQ&BgL)q (S 2 'B'L3q+i-1,

such that 1 < i < q and such that i - 1, sn, ... ,si are not all zero, where s, .. ., si are the

internal degrees in the domain.

By Lemma 2.5, we may represent any homology class of interest by an element E =

Ek Vk, where the Vk E V'+1 are elements of the basis chosen while defining x. We wrote

each Vk as a W-expression ek in various Ukj C Vp:

Vk ek(UkJ) X (I)q+i-1 C FWV,

178

so that dv ek(uk3 ), and defined X(vk) by the formula

X(vk) = kzX(uk)

That each X(Ukj) is a sum of the classes za implies that

0?(X (Vk)) = quC (e re) (X(Uki)).

Taking quC (erp) extracts the part of erp corresponding to the quadratic grading 2 part

of ek, in q2 FW. That is, we may write ek C FWV as

ek = quw(ek) (ukj) + EZ<i<q Ai-1(Oiek)(ukj) + w E FwVp,

where w E FWVP is the quadratic grading = 2 part of ek, if we view quw(ek) E as an

element of FWV via the inclusion F4(n+1)V -+ Fw(n+l)V, and then

9 (X (Vk)) = quC (V (quw(ek))(x(ukj)) + Z i-1(6ek)(X(uki))

= V(quw(e))(x(uk3 )) + OI Z 1( e)(X(uk9))

= ( o S 2 (QWx) 04W + _. 1 0QWX0 e ) (vk).

We were able to discard the application of quC as its argument already has quadratic grading

2. This formula is exactly what we needed to check in order to use the equations in 6.3. 0

Proof of Theorem 13.3. Write L - QU(n)BW(n)X C sLC(n), Z = (n) and W = W(n) (not

W(n + 1)). We only need to show that the diagram of chain complexes

TM (QL Bg L) Tm_1I(S2 (Q4BgL))dg h d he

Nm(Q4 L) > N_1 (S2 ( QfL))

commutes up to homotopy (recall that b reduces filtration by one). The augmentation

maps d are induced by the augmentation of ':

e : (NhNv(QBL) = NvQ" VL -- + NvQ L).

We may understand NQEWL using the pushout square of chain complexes (obtained by

179

applying N,, o Q' to that defining !W):

@yhg(N>1L) F2{zdy} &@xOhg(ZNL) F 2{zx}

0 yChg(N>1L) F2{hy, zdy} -NvQWL

which shows that NvQ49L is the following complex (with differential hy Zdy):

NJvQ4L = Gyehg(N> 1 L) F2 {hy} e GxEhg(ZN.L) IF2 {zx}

We will use the notation

L, := QI"BWX F FF - - Xt

so that we may write the basis elements of NmQ49QUBWX in the form

Z (-1) (h )) and hf(1) (g(o) (hM))

where the hiji 2 are various elements of F ... Fm- 1) Xm, each gil is a W-expression gi1 (hi1i 2 )

in certain of the hiji 2 , and finally, f is some L-expression in the various g2i. For brevity we

will write kf(1) O(h1 ) for either of Zf(1)(9 )(h ) )) and hf(_1)(9 gO)(hM%))'

A chain homotopy 4 : Tm(Q-BL) 4 Nm(S 2 (QWBWX)) -is constructed as follows.

Let D be zero except on NhNv(Q4BWL) N7vQ49QUBWX, where it is defined by

k(-1)((o) (hM i)) Y q~) (gtj (h1T))

This definition makes sense (and yields a non-trivial map) because f is an operator in

QUFW = FE. The chain map d@ + 4d is a sum of three terms:

(a) d o ( : NN(Q4 BW L) -I* Nm(S 2 (QZL)) d Nm_1(S 2 (Qf-L))(b) (1 o dv : NhNv(Q4BW'L) d NhNv _1 (QE BL) A Nm-1(S 2 (QSL )

(c TohN NV 0dh 1QBW)(4)(c) ) o dh :i N -_I(Q4B L) d NhN;_1(QB'L) A Nm-1(S2 (QLL))

We calculate

(d o D)(kf( 9 )(h) d(qu (f)(g (h)'t1 1%2

= quC (f)(gii (h 2 ))

= quC (f)(,E(gjj)(h M))

180

(the last equation holds as we calculate in S2 (Q'LL)), and

(- o d) )(kf (go) (h))

if 'k' stands for 'h',

if 'k' stands for 'z',

(in either case).

By the equation of 3.10, the sum of these two terms is quC(Ef(gii))(h 2 ), which is exactly

the formula for (4w o c)(k g>(hM )'

It remains to show that 4o dh coincides with c®2o Vw. These two maps are only non-zero

on the graded part NhN7,_ (QLBWL) g QCW 2 L of Tm(QLB'L), and an element therein is

a linear combination

K := Ejk kj(CjOkjo 9eL 7 sto k1go Zai2

which satisfies the equation dh(K) = 0, i.e.:

dh(K) = Z k = 0 in Nml-QZWL.ej(fjjO ),0 01 3,0112I

There is a map

Nm-iQZ WL C F2 {hg(Nm-1L)} E F2 {hg(ZNm-1L)} -* Nm_1S 2 QZL

defined on generators using the function

Nm-L C F (FW)mX q S2 (F)ml-X '= S 2QZL.

This map sends di(K) = 0 to

Ej quZ (ej (fji.)) (Sa )9 h - =0,

which by the equation of 3.10, gives an equation in Nm-iS 2Q L:

E quZ (e (,E (fi)))sa 9) h%. )= EY quC (Eu(e) (fi)) (sa.g(9). h(l).

The proof is completed upon noting that the left hand side of this equation equals (E02 0

181

4(kf(gjj)(M ))),

(D(0), 1

=quC (f (gil)) (h )

OS)(K), while the right hand side equals (D o dh)(K). We calculate:

(c o Os) (K) =E (z quG(ej) (s0 oji kfj 9o h! 1 ))

= Zgqu (ei) j i E (9o)g. h()

= E qu (ej (e(fjio))) scj~ 0g Mi-h =1 ) - LHS,

and

dh (K) = Ej

- Ej

D(dh(K)) = E

= Ej

C(ej) (saQikfjj 9() h1 2 ) (in N 1 QWL)

C(ej) (qu (fji) (sj g(9- h( 2 )) (relevant s,3 1 are id)

qu (c(ej)(fjiO)) (sQji0 ~g h =0 1 RHS.

To explain further the third equation, note that N 1 Q49L is spanned by the classes k...,

and none of their degeneracies. Thus, all of the degeneracies sjzo appearing in the second

line that have not already been annihilated during the application of C must be the identity.

Thus, they can be carried harmlessly through to the end of the calculation. El

182

Chapter 14

Calculations of W(n)-cohomology and

the BKSS E2-page

In this section, we will calculate the value of HV(fl)X for certain objects X of W(n) of finite

type. In each subsection, we will write (n) = DX, so that X has underlying vector space

dual to V(n) c VT . In fact, we will reinstate the upper asterisk for linear dualization, writing

DV() :-V(*), and recursively define:

V(k+1) := H7yj)V(*) and V*k+1) := H (k)V(* for k > n.

In this way, for each k > n, ,*+1) is an object of W(k+1), vector space dual to V(k+1) E k+1,

which itself has the structure of an object of MJv(k + 1). Having all of this data will allow us

to draw conclusions about HQ(f) V(n), using, for each k > n, the (k + 1)st composite functor

spectral sequence:

[E (k+1) ]Sk+2,-- -,81 :=( w k 1k3+2, ---,S1 =- H~ ) k +2+Sk+1,8k,- --,Si2:= (H' (k+1)Vi*k+1))t A+2.Si * t.s

The first CFSS, which calculates H* from H* , will appear in 14.5.

14.1. When X C W(n) is one-dimensional and n > 1

Let X = V*) E W(n) be a one dimensional object of W(n), dual to a one-dimensional vector

space V(n) E VT+, with non-zero element v E (V())S 1 Write v* E X ,., for the non-

zero element of X. As every W(n)-operation changes degrees, X is necessarily trivial. We

distinguish two cases: when v is restrictable and when v is not restrictable. Recall that v

is said to be restrictable when its restriction v[21 is defined, i.e. when S,.. . , Si are not all

zero.

183

Proposition 14.1. For each k > n,

V(k) =- FM(k)Fm(k-1) -.-- FM (n+1)(n)

and V* is a trivial object of W(k).

Proof. The proof is by induction, with the case k = n simply our standing assumptions.

If the statement holds for V(k), then Proposition 9.4 shows that the Koszul complex cal-

culating V(k+1) has zero differentials, as V(k) has trivial W(k)-structure, so that V(k+1) =

FMv(k+1)V(k). This has trivial W(k + 1)-structure, by the results of 9.2. E3

Our next step is to calculate, for k > n, the groups:

[E k+1) Sk+2,.sk -.-S.i := (H + ) (+ ) k+2, S .. S1 V )Sk,---,k

The isomorphism shown here follows from the observation that in dimension sk+1 = 0, an

object of W(k + 1) is nothing more than an object of L(k). More precisely, consider the

functor -o : + - V- given by

(Yo)"k,**, := YSk.

Then -o induces a functor -o : W(k + 1) -+ (k), such that, for all Y E W(k + 1):

(FW(k+1)(Y))o a F(k)(Yo) and (QW(k+1)y)o a QZ(k)(y 0 ),

so that (Qw(k+1)Bw(k+1)y)o c (QC(k)B'C(k)yo) for any Y E sW(k + 1), and thus:

Proposition 14.2. Suppose that Y C sW(k + 1), where k > 0. Then

(Hy(k+1)ySk+2,0,Sk,---,Si S (H Yo)Sk)'.-.')1

Returning to the calculation at hand, we may identify a part of the E2 -page with the

Chevalley-Eilenberg-May complex of Appendix A.3:

Proposition 14.3. For each k > n, there is an isomorphism of commutative algebras:

[E(k+1) ]k2,sk,, (D..'(51*)))tk+21Sk.

When v E V(n) is restrictable, DX'(V*) = S(W)[Vik)], the free non-unital commutative

algebra. When v E V(n) is not restrictable, V(k) = F2 {v} is one-dimensional, and DX'(V*/)

is the one-dimensional exterior algebra A(W)[v]. In either case, for each individual value of

184

the grading t, the group

Q [E k+1) Sk+2,0,sk.Si

Sk+2,sk.,S.1

is finite-dimensional.

Proof. The only further observation necessary to prove this isomorphism is that if v c V(n)

is restrictable, every element of the trivial partially restricted Lie algebra V(k) is restrictable,

and that if v E V() is not restrictable, each V(k) is one-dimensional, concentrated in non-

restrictable degree. For the finiteness property, one simply notes that the V(k) have such a

property, and that there is a degree shift in the algebra structure. El

Consider the diagram:

H H U(n)V* H* Hl(n+l)V**(n+1) (n) W(n+2) * (n+1)

H* V*H* V H* V*W(n) (n) I(n+1) (n+1) Wn+2) (n+2)

t Pn Pn+I t Pn+2..

FMhv(n+1) V(n) FMhv(n+2)V(,+l) FMhv(n+3)yn+2)

FMh(n+) V(n+l) fn+i FMh(n+ 2 ) V( 2) f.+2 FMh(n+3) V(n+3)

For each k > n, the map Pk is induced by the inclusion V(k) Hw(k) V C Hw (k) ( (which

exists as V(k) is trivial) and the FMhv(k+l)-operations defined on Hw(k)V(k). (Note that Pk

is a graded map, since the effect of these operations on dimensions is the same in its domain

and codomain.)

The double arrow gk+1, which we use to represent the convergence of the (k + 1)st CFSS,

[E k+l)] -z HyQk)V,*), is in truth shorthand for the function

[E k+)Sk+2.... .--- Si - + [EoH2(k)V( k+2)..,

so that gk+1 may only be defined on the permanent cycles within [E k+l)], and lands in the

associated graded of Hw(k)(

Similarly, we employ the double arrow fk+1 as shorthand for the function of Theorem

8.15, which is defined on the entirety of F kMh(k+2)FMV(k+ 2)V(k+l), but whose true codomain

is the graded object Eo(FMh(k+l)V(k+l)) associated with the target filtration defined in

Theorem 8.15.

Theorem 14.4. For each k > n, im(pk+1) consists of permanent cycles and Pk preserves the

target filtrations, so that it is possible to form the composites gk+1 0 Pk+1 and Eo(pk) o fk+1-

These composites are equal, and moreover, Pk is an isomorphism. In particular, for k > n,

the (k + 1)" CFSS collapses at E2 .

185

Before giving the proof, we remark that in some dimensions we already have enough

information to see that Pk is an isomorphism. Specifically, we prove:

Proposition 14.5. For k > n, Pk is an isomorphism in dimension sk = 0:

Pk: (FMh(k+1)FMv(k ) k))Sk+1,Osk-1... .3l ) Sk+1,0,sk-1,...,S1

Proof. In this dimension, Pk factors as

(FMh(k 1)FMv(k+1)y k+1k1sk-1 .S1 = (FMh(k+1)FMv(k+1)yVg))Sk+1,OSk-1. --. 1

= (FMh(k+1)V )Sk+1,Oik-1 ,.-,i1

m (DX/((V( )))k+1,Osk-1,-.,Si

S(H' (k)(k) ts+,,k1.-s

Here, we are viewing V 0 the subspace of V(k) in degree sk = 0, as an object of Vk+1

in order to apply FMv(k+1). The inclusion FMh(k+1)FMv(k+1)yo C FMh(k+l)FMv(k+l)V(k)"(k) - Vk

restricts to the identity in degree sk = 0, explaining the first equation. The second equation

is similar: any non-trivial MV(k + 1)-operation lands outside degree Sk 0. The first

isomorphism follows from Corollary 8.11, which ensures that FMh(k+l)y) is a quotient of"(k)

the polynomial algebra on V), and indeed, the same quotient as DX'(D(V))). The second

isomorphism is Proposition 14.3, since (V* ))o -E

Proof of Theorem 14.4. For each k > n, we will use the diagram

. de .com posite --...

FMh(k1)FMv (k+1) g+1 W(FMv(k+1) .(k) .. FMv(k1) V(k)

where W(FMV(k+l)V(k)) is the object introduced in the proof of Theorem 8.15, so that there

is a quotient map

W(FM-(k+1) V(k)) -*- FMh(k 2)FM'(k+ 2)FMV(k+1) V(k)

Here, the maps ji, j 2 are the evident inclusions of generators, while the maps iO, ii, i 2 are

the inclusions arising because V(k) is trivial.

We may define

C : (Pk 0 f k+ 1 0 F1 :F (k)V(k) -_+ Hw (k) (k'

186

without the need to pass to any associated graded objects. By construction of fk+1, c is

induced by the inclusion io and the M,(k + 1)-structure of HN(k)(k)'

The edge composite is the composite of a surjection, a monomorphism mI, and an

isomorphism m2 (with inverse ii):

HwQ(k) V(* -* [EOH-(k)V(*] [Ek+) 0 - [E k+1) H((k)V(k).

Moreover, c is a section of the edge composite, since both maps are compatible with the

MY (k+ 1)-structures (Proposition 12.9), and their composite is the identity on the generating

subspace V(k) of FMv(k+l)V(k). In particular, the edge composite is a surjection, so that m,

is an isomorphism. That is, every class in im(ii) is a permanent cycle. Singer's work

(c.f. 13.1) then shows that im(Pk+1) consists of permanent cycles, as permanent cycles are

preserved by the FMhv(k+ 2)-operations on Ek+l)].

Any section of H* (k)(/ - [E, ] [E k+l)lo will realize, up to filtration, the

restriction of 9k+1 to E, ]C [E00 ], so we choose

0* m* '- rmv(k+l)T KJ* *E2,(k+l) -+ H * (k) V(k) - H(k) HI(l)V(k)-

In particular, gk+1 0 Pk+1 0 Ji = 9k+1 0 il C 0 m2 0 i = c, up to filtration. More precisely,

9k+1 0 Pk+1 0 ji equals the composite

FM-(k+1)V(k) -c H,(k)V/*) -- > [E0 HW(k)V(k)1 0.

Now the target filtrations on the domain and codomain of Pk are induced by the filtrations on

the domain and codomain of Pk+1 by cohomological dimension sk+2, and Pk+1 is a graded

map. Thus, for any w G FPW(FMv(k+l)V(k)), we must see that pk(fk+l(w)) coincides

with gk+l(pk+l(w)) modulo FP+1H~Q,) )V as this will prove both that Pk preserves target

filtrations and that 9k+1 0 Pk+1 = Eo(pk) 0 fk+1. However, this coincidence follows from the

fact that c = 9k+1 0 Pk+1 0 ii, as W(FMv(k+l)V(k)) is generated by im(ji) under FMhv(k+ 2)_

operations, and the definition of fk+1 is modelled on the interaction of 9k+1 with these

operations, as studied by Singer (c.f. 13.1).

What remains is to show that the maps Pk are isomorphisms. Suppose that

X(k) C [E2k)]+1 -' = (H-(k) V k 1 ,.,t 1

Now X(k) is detected by some permanent cycle X(k+l) E [E k+l)], which is detected by some

187

permanent cycle X(k+2) C [E k+ 2)], and so on, giving a sequence of elements

X(r) E [E )]r5+1-I, = (H(r) ( )) . for r > k,

where s+ 1 + s; = s~ and s = for 1 k.

We will say that X(k) has iterated filtration at least (s s s ,k.2. .) whenever a sequence

of such classes X(r) exists, and partially order the set of possible iterated filtrations lexico-

graphically. Then x(r) only determines X(k) modulo elements of E2,(k) of higher iterated

filtration.

Simply because these gradings are always non-negative, it is inevitable that sr = 0 for

some r > k, so that by Proposition 14.5, x(r) = PrY(r) for some y(r) E FMh(r+l)FMv(r+l)V(r).

Moreover, one only needs to examine finitely many sequences of gradings, each of the form

(sr+ , 0, s_1 ,* s+ 1 , sk, ... , s) where sk+1 = sr+1 + s_ 1 + sr-2 + - + s+I

This, along with Proposition 14.3, shows that (H(k) ))+1." is finite dimensional for

each given value of t.

By the commutativity established above, X(k) Pkk+1 ... fr-lfr(Y(r)), modulo higher

iterated filtration. As this congruence holds in a group which is finite dimensional for each

given t, this establishes the surjectivity of pk, and that every one of the spectral sequences

is degenerate. Thus, we have shown that all of the maps gk are in fact isomorphisms, or

rather that in the following commuting square, for any k > n, gk+1 is an isomorphism:

E0 V 9k+1 Hu(k) V*

Eo (pk) fk Pk+1

[EOFMh(k+l) k+(k 1)] 1c FMh (k+2) FMv(k+2)FMv(k+1) V(k)

For each k, Pk is injective if and only if EO(pk) is injective. This holds by repeated application

of the snake lemma, using the fact that Pk is surjective, and the observation that for any

given value of the grading t, the group (FMh(k+l)V(k+l))t is finite dimensional, so that the

filtrations of both the domain and codomain of Pk are eventually zero in each degree t. More

specifically,

Pk : (FMh(k+l) V(kl))Sk+1,---S (H-4 (k) V)) l.si

is injective if and only if

EO(pk): [EoFMh(k+1) V2"k+2 8 k1,Sk,---,41 -)'k+218k+VIk,-,81

is injective whenever s42 + s'l = sk+1. As in the argument for surjectivity, in order to

188

check that all the Pk are injective, we now only need to check that every map

(Fh (r+')FNv(r+1)v(r)) t 12sM1Si V ( rt

is injective, which is part of Proposition 14.5. El

14.2. A Kiinneth Theorem for W(n)-cohomology

This is an opportune moment to prove:

Theorem 14.6. Suppose that X,Y E W(n) are of finite type, with n > 0. Then

H- ) (X x Y) H n (X) L Hi tn(Y)

where the coproduct is of non-unital commutative algebras.

Proof. This follows from the Kiinneth Theorem (6.15) adapted to s (k), and the observation

that HM(k)(Z x Z') 2 H (k) Z x H2(k) Z', using the techniques of the proof of Theorem

14.4.

Theorems 14.4 and 14.6 together imply:

0

Corollary 14.7. For n > 1, the category Mhv(n + 1) is the category HW(n) of W(n)-H*-

algebras.

14.3. A two-dimensional example in W(2)

In this section, we suppose that T > 1, and let X = V*) E W(2) be the two-dimensional

object of W(2) spanned by non-zero classes

E (V*2)) 1 and v* E (V(*))2+1

such that v* = v3Ao = (v*)[2], and with all other operations trivial.

Proposition 14.8. For all k > 2, V* is two-dimensional, spanned by

v E (V)) ..,0,1 and v* E (V

with v* = (v*)[2] the only non-trivial operation.

Proof. An induction as in the proof of Proposition 14.1, using the fact that at each stage,

the only non-trivial A-operation is a top operation, and thus does not yield a differential in

189

K*(k)V(k). One also uses Propositions 9.9, 9.11 and 9.12 to calculate the W(k + 1)-structure

of V(k+1) at each stage. FI

Proposition 14.9. For each k > 2,

(H.(k+1)S(1)i ( ) (k) ' (S(%')[v ] i AQ(t)[vo]),k 2 ,Sk.

These groups are zero unless sk = = 2 = 0.

Proof. One performs this calculation in the Chevalley-Eilenberg-May complex DX'(V*k

which by Proposition A.9 is the differential graded algebra F2[vo, v1] with differential

d(vo) - ([2/W) 2 = 0, d(vi) = ([V )2 = v 2

By a greatly simplified version of the proof of Theorem 14.4:

Corollary 14.10. For each k > 2, (E2 ,(k+l)) k+2,Sk+,---,Si is zero unless sk+1 = ... 2 = 0,

so that the spectral sequence E2,(k+l) ==-> H (k)*) collapses, and in particular,

HO( S(i)[v2] H A(W)[vo].

14.4. An infinite-dimensional example in W(1)

In this section, we suppose that S, T > 1, and let X = * W(1) be the infinite dimen-

sional object of W(1) spanned by non-zero classes

Vj (V*1)2j(T 1>.4 for j > 0,

such that v.7+1 = v>A1 for j > 0, and in which all other operations are trivial.

Proposition 14.11. The Koszul complex K V* has basis

{Sqj*(vj) I j > 0, J is Sq-admissible, _M(J) <; S + j and 1 J}

and all differentials zero except for:

Sqjet-,i2,2)*(v*) Sq' ,.,i2)*(V

Proof. The basis given for the Koszul complex is just a reading of Proposition 9.4, but we

190

E]

must think a little about the differentials. As Al is the only non-zero operation:

d(Sq%*(vj)) = 2 SqVk'.(ke,...,k2,2) J

(ke,.,k2 ) Sq-admis.

Consider a sequence (ke, ... , k 2 , 2) corresponding to a summand of this formula. Supposing

that f > 2 and Sqk2Sq2 is not Sq-admissible, it follows that k 2 is either 3 or 2, so that Sqk2Sq2

is either zero or SqlSq,. As J does not contain 1, and the two-sided ideal in A generated

by Sqhl is spanned by those admissible sequences ending in Sq1, it cannot happen that

(k. .. , k2 , 2) 4 J. Thus, the only summand appearing is that in which (k, ... , k2 , 2) = J,

confirming our description of the differential. El

Proposition 14.12. When S > 2, V* := H U()V* is the subquotient(2) * (1)

F 2 {SqJ*(v;) j 0, J is Sq-admissible, m(J) S + j and 1, 2 V J

F 2 {SqJ*(v*) j 1, J is Sq-admissible, m(J) 5 S+ j and 1, 2, 3 V J

KU(l)V* M2of K V*. Equivalently, V(2) is the subquotient of F 2V(2) in which we restrict to the

sub-Mv(2)-object generated by the elements

{vo, Sq2vi, Sq3vi, Sq2v 2 , Sq3v2, Sq2v3, Sqv 3 ,. .

and in which we set Sq~vj to zero for all j > 0. As an object of W(2), V*) is trivial.

Proposition 14.13. When S = 1, V* := H U()V* is the subquotient(2) (1)

F2 {SqJ*(v*) j 0, J is Sq-admissible, m(J) S + j and 1, 2 J }F2 {Sqj*(v*) j 2, J is Sq-admissible, m(J) S+ j and 1, 2, 3 V J

of K*V* . Equivalently, V(2) is the subquotient of FM( 2 )Vil in which we restrict to the

sub-MY(2)-object generated by the elements

{vo, vI, Sq~v2, Sq~v2, Sq~v3, Sqiv 3 ,. .

and in which we set Sq~v to zero for all j > 1. As an object of W(2), V*) admits a single

non-zero operation, A0 : v* - v*, and so decomposes as the direct sum of F2 {v*, v*} with a

trivial object D(V')), dual to V'(), the subquotient of FMV(2 )V(1) in which we restrict to the

sub-M,(2)-object generated by {Sq~v2, Sqiv2, Sqjv3, SqVv3, .. .} and set Sqv to zero for all

j 2.

191

Proof of Propositions 14.12 and 14.13. For any S > 1, taking the homology of this differen-

tial provides the formula for V%, and dualizing provides that for V(2). In order to determine

V(% as an object of W(2), note first that 9.9 and 9.11 show that all operations are zero except

perhaps for A 0 . Consider the operation A 0 applied to a cycle of the form Sq* (v*) E K(') V*)

with J = 0 (so that 1, 2 J). As J ends in an integer no less than 3, and as A, is the only

non-zero operation in V*, the second part of Proposition 9.12 implies that Sqv*(v*)Ao = 0.

In the case J = 0, Proposition 9.12 states that (Sq0*(v ))Ao c V* is represented by

(Sq0*(v AS+j)), which is zero unless j = 0 and S = 1. Thus the only non-zero operation on

V*) is v*Ao = v* in the case S = 1. D

Theorem 14.14. The spectral sequence H V 2 - H (1)V*) collapses, with

E FMh (3) FM()V(2), if S > 2;[ EF ] = H( 2)1% ( 2) iS2)2

2 FMh( 3 )FMv( 3 )V) j S()[v] L A(c)[vo], if S = 1.

Proof. The calculations of [E 2 ] follow from Theorems 14.4 and 14.6, Propositions 14.8,

14.12 and 14.13 and Corollary 14.10. What remains is to prove the collapsing result in each

case.

Suppose that S > 2. The first point is to observe that the generators vo and Sq3vj

(j 1) of V(2) under Mv(2)-operations are all permanent cycles in (H (2 )V( 2))2**. For

v 0 E [E]2 ) ] S, this is obvious. It is less obvious for Sqiv (j > 1), which has only one

opportunity to support a differential:

S 3 Ej(2)]0,1,2+S+j d2 E(2) ]2,0,2 + S+jqVo E [E2 2j+1(T+1)-l14 [E2 2j+l(T+1)-l'

Fortunately, this target group is zero, due to the constraint that S2 = 0. To see this, note

that this group is spanned by three-fold products of classes in [E 2)]20*, namely:

VilVi V c- E (2)]2,0,3S+i1+i2+303102033E [2 (2i1+2i2+2i3)(T+1)-1'

and if this target group is non-zero, these indices must coincide. In order that 2 j+1 equals

2il + 232 + 23 it must happen that Jij2, j equal j, j - 1, j - 1 (in some order), but then

2+ S+j =3 - -ji +J 2 + j 3 implies that S+j=2. This is impossibl, as S 2 2 and j 1.

Next, we can derive that Sqvvj is a permanent cycle for all Sq-admissible J and j > 0

such that J has final entry 3 when j > 0. For this, we will use the following commuting

192

diagram from Proposition 12.9:

(H, , V~~2s sI (H (I V*) 2t~+l -

ledge hom ledge hom

[E (2]O,S2,S1 [E (2] ,2+1I,S1 +i-1

(V(2))t2s a (V(2)) 2t s+ -

As we have shown that the classes vo and Sqiv (j > 1) are all permanent cycles, they are

in the image of the edge homomorphism. Then this diagram shows that all of V(2) is in

the image of the edge homomorphism, so that every element of V(2) is a permanent cycle.

Finally, as E 2 is (freely) generated by V(2) under the Mhv(3)-operations, and we understand

how these operations interact with the differential, this shows that the spectral sequence

collapses.

Suppose instead that S = 1. Then rather that having generators vo and Sqiv, (j 1) as

before, E2 has generators vo, vi and Sqvj (j 2). Note that Sqivi = 0 when S = 1. That

V E [E2 )4T 3 cannot support differentials is obvious, while for j 2, the same degree

argument as before shows that Sq vv is also a permanent cycle. The same argument with

the edge homomorphism shows that every element of V) is a permanent cycle, so E2 is again

generated by permanent cycles under the Mhv(3)-operations, completing the proof. 0

Corollary 14.15. If S > 2, then HQ()V*) is isomorphic, as a vector space in V2, to the

subquotient of FMh( 2)FMv( 2 )V(1) generated by the elements

{vO, Sq~vi, Sqivi, Sq~v 2 , Sqiv 2 , Sqv 3 , Sqvv3 , ...

and subject to relations generated by Sq v = 0 for all j > 0. Under Mhv(2)-operations,

H (I) V* is generated by vo, SqV, Sqv2, etc.

If S = 1, then HQ (1 V * Vi is isomorphic, as a vector space in Vt, to the commutative

algebra coproduct

subquo U S(e)[v ] Li A(e)[vo],

where v 2 (H (1) V ) (T_l, and subquo is the subquotient of FMh(2)FMV(2 )V(I) gener-

ated by the elements

{Sq~v 2 , Sqiv 2 , Sq v3, SqV 3 , ..- }

and subject to relations generated by Sq$vj = 0 for all j > 2. Under this isomorphism,

Hw(1 ) V(0 is generated by vo, vi, Sqiv2, Sqiv3 , et cetera, under the Mhv(2)-operations.

193

Proof. Suppose first that S > 2. Consider the elements

1S 3S(V(2)* 31,S+i+2 ( )V0 G (V(2 ))OT S, Sqvo E (V(2)) 2 'S+ 1 (j ;> 2), and Sqivi E (V(2)2 (+

These elements span (V(2 ))2'* and (V(2 ))*'*, and can all be distinguished by their internal

degrees, so the restrictions

(H V( '-- [E(2]0* = (V( 2))2'*, (H)()V())'* - [E ]O'I'*=

of the edge composite (c.f. Proposition 12.9) are isomorphisms. We write

h : (V(2))O*'* G (V(2))*I'* ->_ H' (J) (*

for the injection obtained by adding their inverse maps. Use the basis of V(2 ) arising from

Propositions 14.12 and 8.8 to extend h to a vector space map H : V(2) -+ H (I)V) by the

rule H(Sqvx) = Sq?1H(x). Although H is not a map in Mv(2), it does induce the vector

space isomorphism required for the proposition.

Suppose instead that S = 1. The same argument produces a map subquo -+ H V-

The difference is that we must find candidates for v, and vo in H(,*)V*. We send vo to

the unique non-zero element of (H,1)V())0 )1 and v2 to the unique non-zero element of

(Hw (1)V 14(T+1-1'

14.5. The Bousfield-Kan E2-page for a sphere

Let X = V E W(O) be a one dimensional object of W(O), dual to a one-dimensional vector

space V() E V+, with non-zero element z E (V(O))T. Write t* E XT for the non-zero element

of X.

As every W(O)-operation changes degrees, X is necessarily trivial. Moreover, it is

quadratically graded, by setting z* E qiXT. By Proposition 12.8, the first CFSS will admit

a quadratic grading.

Recall the function 'T : adm+ (A, T) -- > adm+(A, T) of 9.2. In view of the strict

inequality derived during the proof of Lemma 9.10, it need not be true that I is of the

form T'_ (ij+1 ,... , i1) whenever I = (ie,. . . , ii) E adm+(A, T) satisfies (ij+1, ... ,I 1 )

TT (ij,... , ii). Nevertheless, we may use TT to decompose adm+(A, T). Define:

admi (A, T) := adm+(A, T) \ im(TT : adm+(A, T) -+ adm+(A, T)),

the set of sequences in adm+(A, T) not in the image of TT, so that we may decompose

194

adm+(A, T) as the disjoint union

adm+(A, T) = LiIEadm+ (A,T)

Proposition 14.16. V: H* (0)V* has basis {* } Li {)J*t* I I E adm+(A, T)} and all

W(1)-operations trivial except for A 1 , which is defined (only when f(I) > 1) by

51*Z* A+ JV* it*.

Thus, as an object of W(1), V1*' decomposes as a direct sum

F2 {Z*} e QIEadm+ (A,T)

F, oJv*(t*Aj) I j 0}.

Proof. The basis of the Koszul complex was described in Proposition 9.2, and the Koszul

differential is zero as X is trivial. The A-operations were calculated in Proposition 9.11. El

Now we have put considerable effort into calculating HQ(,) of each summand in this

decomposition: Theorem 14.4 proves that

H 7(1 (F2{W*) -_ F lh(2)Fm ( F2{IZ}) _- A(% )(t),

while Propositions 14.12 and 14.13 calculate

Hw(1 ) F2 {(3J*Z*)AI j 2 0 for I c admr (A, T).

With a view to calculating the first CFSS, we catalogue a collection of generators of [E)]

under the Mhv(2)-operations. The fundamental class z E qi[El)]0O is an exterior generator

(arising in Theorem 14.4). Moreover, for all I E admir(A, T), there are further generators,

arising in Corollary 14.15:

Rjz E q2fI[E(1] I+

Sqv 4 z Ez- q l+e + [E ]$ + + ++1)-11,2+ +j( iSiz)2 2 [E 2 2i+1(T+nfI+ 1)-1

(14.1)

(when j > 1, but not j = fI = 1), (14.2)

(when I = 1), (14.3)

where they are referred to as vo, Sqiv and v, respectively. Note that this final generator,

(Ovz)2 , has the same degrees as the generator Sq 3pz that is missing when j = I = 1.

195

IT1, T , ... .

Theorem 14.17. The first CFSS collapses at E 2 :

[E2]= H* 1/)V*O = HlQ(O)VO).

Proof. The fundamental class is a permanent cycle, so to prove that the spectral sequence

collapses, it is enough to show that no classes

x q2 E(1) or y C q2 eI+ 1,2++jz Eq~t (2 T+nI+iI oryEq1uy[2 12i+1(T+nl+f1+ 1)-1

can support a differential, for I a non-empty 6-admissible sequence.

To see this, all one needs to have learned about the entire E2 -page is that it is a sub-

quotient (in which z2 = 0) of the polynomial algebra on symbols

Sq A BSq 6v G q2 (A+,B+eC[E1)]eB+nA,2 A(nB-eB+C)h v C E[2 2A+eB(T+nC+fC+1)-1'

in which B is Sq-admissible, B does not contain 1 or 2, if C is empty then so is B, and if B

is empty then so is A. These conditions imply that nB - 2fB > 2 B - 1.

If for r > 2 there is a differential d, supported by y, then dry must be a sum of products

of N > 1 such classes. The generic such monomial may be written as:

rN qAkS Bk6 z E2A kk[(1)] (fBk+nAk)+N-1,J: 21Ak (nBk -fBk +fCO)Hk=1 Sqh Sqv q( 2 Ak+eBk+ Ck2 -1+E 2^k+lk(T+nCk+tCk+1)

in which kCk = 0 for at most one k. We derive the following constraints:

Z(fBk + nAk) > 4 - N, (14.4)

log2 (N) + N EZk [fAk +tiBk + LCk] 1I + j, (14.5)

4+ I + j = k [fBk +nAk] + N -1A+k [2Ak(nBk - fBk +eCk)] , (14.6)

log2 (N) Zk (( 2eAk - 1)Ck + [(2eAk(nBk -eBk) - 1 Bk) - 1(iAk)]) . (14.7)

The inequality (14.4) is just the requirement that r > 2, while (14.5) results from the

observation that d, preserves the quadratic grading and the convexity of the exponential

function. Equation (14.6) holds since the total degree of the differential is one, and (14.7) is

derived by rearranging the sum of (14.4), (14.5) and (14.6). (14.7) is a very strong inequality,

since the expression 2 Ak(nBk - fBk) - k!Bk is at least 2 eBk - and nB - Bk 2 if

fBk # 0. Thus, in (14.7), each expression in square brackets is always non-negative, is at1 1 ~~~1 -r / -

least 2 - y when tBk # 0, and exceeds 2- if cBy 2 or tAk #0.

When N = 1 or N = 3, log2 (N) < 2 - 1, so that (14.7) implies that fBk = 0 for all

k, violating (14.4). When N < 2, log2 (N) 2 - y, so that (14.7) implies that fBk = 0N' a(1.)ipista k 0

196

for at most one k, with fBk = 1, violating (14.4). When N > 4, all but at most one of the

summands (2eAk - -)fCk in (14.7) is at least 3, and as !(N - 1) log2 (N) when N > 4,

(14.7) is violated. Thus y E E2 is a permanent cycle.

Performing the same calculations for drx, we find that the inequality (14.7) is unchanged,

while (14.4) is replaced by

Z(LBk + nAk) 3 - N. (14.8)

The argument is unchanged when N =1 or N > 4, while if 2 < N < 3 we may still draw

the same conclusions from (14.7). When N = 2, we may assume that LB 1 = 1 and LB 2 = 0,

and although (14.8) is not violated, (14.7) is violated as C, = 0. When N = 3, we must

have LCk = 0 for each k, and the following equations must be satisfied

I-i =LC1 +LC 2 +LC 3 , 2 ei = 2e01 + 2 0 2 + 2 ec3

As in the proof of Theorem 14.14, these equations imply that LC1 , fC2, LC3 equal LI- 1, fI-

2, LI - 2, in some order. The first equation then implies that LI = 2, implying that ECk = 0

for more than one k, which we have prohibited. Thus x E E2 is a permanent cycle. E

This theorem has the following corollary, stated in the following form due to the potential

for hidden extensions, which we have not ruled out:

Corollary 14.18. Suppose that X = SWO' for T > 1. Then the BKSS E2-page [E2X] is

isomorphic, as a vector space in V1, to the Mh (1)-subquotient of FMh(1)FMv()f{z} generated

by the fundamental class z and the elements

{Jv, SqV6 t, Sq 8lz, Sqljgj, , .} for I e admir(A, T),

and subject to relations generated under MJh(1)-operations by

{Sq~6jz, Sq 6z, Sq ,,, Sq vZ, .. } for I E adm' (/,T).

Proof. This follows from the collapsing of the first CFSS, our knowledge of the generators z

and (14.1)-(14.3) of [EPl)1, and a few observations in the low-dimensional cases.

When lI = 0: in FMh(1)FMv(1){Z}, by unstableness of the horizontal Steenrod operations,

Sq2Z = 0, Sq32 = 0 and z2 - Sq1' = 0, so that z contributes no more to this subquotient

than it did as an exterior generator of [E,']1.

When LI = 1: in FMh(1)FMv(z){l}, the generators (14.3) satisfy (6vTI )2 = Sq3 Iogz, and

taking the quotient by Sq~~2i2 ensures that these generators produce no more material

197

in Flh(l)FM-(l){} than the polynomial algebras arising in the S = 1 case of Corollary

14.15.

14.6. An alternative Bousfield-Kan E1-page

We will now suggest an alternative E1-page for the BKSS for a sphere X =SO"2 for T > 1,

which will be motivated by the conjectures and calculations of 16. Define:

adm(A>1, s) := {J I J a Sq-admissible sequence with m(I) < s + 1, 1 J};

admirr(A>1, s) := {J I J a Sq-admissible sequence with e(I) s, 1 J};

adm(A, T) := {I I a 6-admissible sequence with i7T(I) 5 T}.

The difference between adm(A, T) and adm+(A, T) is just that we have removed the require-

ment that I be non-empty. The following lemma explains the sense in which admirr(A>1 , s)

is the subset of irreducible sequences in adm(A>i, s).

Lemma 14.19. There is an injective function 3t : adm(A>1, s) -- adm(A>i, s) given by

C5J = (je,..., ji) - (s + nJ+ 1, je, ..., ji),

and moreover, admirr (A>1, s) = adm(A>i, s) \ im(G,), so that

adm(A>1, s) = UJEadmi (A>1,s) {GSJ, * J '

The proof is similar to that of Lemma 9.10, but the outcome is a little different. Indeed,

Lemma 14.19 shows that if a Mh(1)-expression SqhX contains a top Steenrod operation,

then all of the Steenrod operations following it are also top operations.

Define

SF N qJA6V Ik E adm(A, T), Jk G adm(A > 1, k)[E(X] := F2 ]L k- kSqhUIO (Jk zJkk) ~ (14.9)

and define a differential on [E'X] by:

(14.10) setting diz = 0;

(14.11) requiring that di distributes across the monomials in (14.9) according to the Leibniz

rule;

(14.12) requiring, for x E [E'X]', that Sq'+2 x = xdix and Sqz= 0 for j > s + 2;

(14.13) requiring, for x c [E'X]', that dlSqhx = Sq2 dix;

198

Jy dix, if 2 < i < t;(14.14) requiring, for x E [E)X]8, that dilzx = i

lj dix + SqiX if 2 < i = t;

(14.15) enforcing the equation 6,'Sq = 0;

(14.16) enforcing the Sq-Adem relations and the identity Sqh = 0;

(14.17) whenever a summand in the image of d, violates the requirement that the factors

SqJk oz be unique, applying the unstableness condition

(Sq41'j t)2 = Sq6enkJk 6z.

Note that (14.12), (14.16) and (14.17) imply that z2 = 0 and Sq = 0. The key point is

that we do not want the differential to be determined by manipulations such as:

di (Sq h( 22 ,10,5,2)2) "=" d1 ((6( 22 ,10,5 ,2)z)2) "=" 2(522,10,5,2)2)(di6( 22 ,1 0 ,5 ,2)t) = 0,

which is why the phrasing of (14.11) and (14.17) is so restrictive. Indeed, when we define

in 11 operations on the Bousfield-Kan spectral sequence, the top Steenrod operation will

not equal the product-square at El, and we are mimicking this behaviour in the present

definition.

Let us calculate the proposed differential applied to a generator Sqhj6 of [E1X] with

I i 0. Suppose that I = (iu,. .. ,ii), with 6Y' acting as a top operation at precisely the

indices a = an,... , a1 . Then we calculate,

diSqhj I = Sqhdi6jz

M h~= Sq ,..i.>7n j)Sq 6 i .. i

SqjSq 26v P SI a top operation, 2, 3 J, f I > 2

0, otherwise.

The first equation holds by (14.13), and the second holds by (14.10) and (14.14). To explain

the third equation, all of the n summands vanish by (14.15), except perhaps for the m = n

summand, which need not vanish when a, = it. Even this summand may still vanish, as

(14.16) implies that SqjSq2 vanishes unless it is already Sq-admissible.

Although the definition of this complex seemed complicated, the differential ends up

being quite simple. Indeed, one deduces that, writing J = (ieg, ... , ji):

if J C adm(A>i, s) is non-empty and 2, 3 V J, then (jei,..., ji, 2) C admirr(A>i, s - 1),

199

and from this, we conclude the following. If J is non-empty:

Sqj65> is a cycle if and only if I C admiir(A, T) or J contains 2 or 3, (14.18)

and if J is empty but I is non-empty:

6'z is a cycle if and only if I E adm+r(A, T). (14.19)

(Recall that adm r(A, T) contains all of the length one sequences (i) for 2 is a cycle if and only if I = 0 or I c admir(A, T) or J contains 2 or 3. (14.20)

The determination of the homology of [E'X], which we denote by [E'X] will follow from a

generalization of this calculation made in 16.2, in particular Proposition 16.3. While the

calculations in 16.2 are contingent on Conjectures 1 and 2, the statements are independent

of these conjectures insofar as they apply to [E)X]. As a result, we can state the following:

Corollary 14.20 (of Proposition 16.3). The homology [E'X] := H,([EGX]) is isomorphic,

as a vector space, to [E2 J] as calculated in Corollary 14.18.

Proof. The isomorphism of vector spaces [E'X] -+ [E2X] sends the class of one of the cycles

Sqj6z of (14.20) to the element Sqj6z of the subquotient of FMh(1)FM'(1){} identified in

Corollary 14.18. Proposition 16.3 provides a basis of H,[E'X] which can be compared

directly with that of the subquotient. E

200

Chapter 15

A May-Koszul spectral sequence for

W(0)-cohomology

15.1. The quadratic filtration and resulting spectral sequence

Suppose that X E sW(n) for n > 0, and write QBX E sV+ for the simplicial bar construction

calculating H* X:

(Qw(n)B W(n)X), , (Fw(n))sX,.

We may view the vector space UW()X as being quadratically graded, concentrated in

quadratic grading 1, and as explained in 12.4, the monad Fw(n) may be promoted to a

monad on qV+, so that QBX is quadratically graded in each simplicial degree individually.

We derive from these gradings the quadratic filtration, the following increasing filtration

of N*QBX E ch+V+:

FmN*QBX =k@m;qkN*QBX.

This definition is the direct analogue of Priddy's definition [47]. It appears to be impossible

to use his techniques to calculate, say, HQ(O)H*OmSfOm directly, as the bar construction in

W(0) grows so much faster than the bar construction in a category of modules, and the

resulting spectral sequence is not degenerate. Nonetheless, the quadratic filtration is finite

in each internal degree:

Lemma 15.1. Suppose that n > 0, X E sW(n), and k > 0. Then for any sk, ... ,s1 > 0

and t > 1,

(F2tlN*QBX), = (N*QBX).

Proof. This follows from the observation that every possible unary (resp. quadratic) op-

eration increases t by at least one and doubles quadratic gradings (resp. adds quadratic

gradings). It is obvious in dimension t = 1, as there can have been no non-trivial operations

201

applied in this dimension (the grading t is always non-negative). The full statement follows

by induction on t. E

Moreover, there is an isomorphism

[F ON*QBX] - NQ'(")BW(n)KW(n)Uw(n)X

of chain complexes, so that:

Proposition 15.2. The cohomotopy spectral sequence of the quadratic filtration is a strongly

convergent spectral sequence, the May-Koszul spectral sequence:

[E.KN1QBX] ,. qm(H1 (f)KW(n)UW(n)X)s.'' = (H n(f)X)'.t

If ,r*X is of finite type, the E1 -page may be rewritten as:

[EIKN*QBX]mS,., q (FHW(f)D(,rX)),fl.

which reduces when n > 1 to:

[E KN*QBX]MSn,..,s1 q(FMhv(n+1)D (r*X))n'..'S.

Notes that all of the spectral sequence operations defined in 8 respect the quadratic filtration

- the unary operations double quadratic filtrations while the pairing operations sum then.

We leave it to the interested reader to derive the resulting theory of operations in the May-

Koszul spectral sequence from this fact, for any n > 0.

15.2. A vanishing line on the Bousfield-Kan E2-page

It is possible to obtain by the following method a vanishing line of slope 4/5 whenever 7rjX

is of finite type. In the interest of brevity however, we prove only the following:

Theorem 15.3. If X E stom is connected (with 7rX not necessarily of finite type) then

the BKSS admits a vanishing line on E 2 of slope 1 and intercept 0:

[E2 X]j = 0 whenever s > 1 - (t - s).

Proof. We will prove that the right derived functors

((RS PrHWom-coalg)W)t

202

have such a vanishing line for any W E HWom-coalg with WO = 0. Any such W is the

union of its finite-dimensional subobjects, as all of the structure maps in W(0) increase the

degree t, so it is enough to prove this Proposition for finite-dimensional W. Then, by passing

to duals, it is enough to produce a vanishing line in the isomorphic vector space

H (0) DW.

This group is calculated by the May-Koszul spectral sequence whose El-page is given by

[E MK]'s - qm(Hv(O)KW(O)Uw(O)DW)8.

Now Kw(O)Uw(o)DW decomposes as a product (for various T > 1):

W(0),T1 W(0),TNK6 x ...x K0

so if we can prove that

(H*)s := (HQ(0 ) K6w(O),T )s=0 whenever s > t - s

the same will be true for H(O)DW by Theorem 14.6. However, we have already calculated

these groups in Corollary 14.18, and found that (H*)' is spanned by the image of z E (H*)r

under various MY(1)- and Mh(1)-operations. All of these operations preserve the half-plane

specified by s < t - s. LI

203

204

Chapter 16

The Bousfield-Kan spectral sequence

for SomT

For any T > 1, let X =S m, so that we may write [ErX] for the Bousfield-Kan spectral

sequence of the sphere S m. In this chapter we will give conjectures which will allows us

to construct a complete system of differentials in [EX], that would explain the convergence

of [E2X] (whose underlying vector space was calculated in Corollary 14.18) to

(S"Om) ~_- A(W)[6z I E adme(A, T)].

Here z C 7rT(S om) is the fundamental class (c.f. Proposition 5.6), and we write

adme(A, T) := {I I is 6-admissible, e(I) < T}.

16.1. Some conjectures on the Ei-level structure

In order to construct all of the differentials needed, we will assume from this point on:

Conjecture 1. It is possible to modify the definitions of the spectral sequence operations A,

Sqh and Ji defined in 11.3 in order that the Sq-Adem relations and the relations JySqj = 0

hold on E1 (without compromising the existing properties of these operations summarized in

Proposition 11.2 and Corollaries 11.3-11.7).

From now on, we will replace the operations defined in 11.3 with their conjectural counter-

part (without change of notation).

Recall the alternative Bousfield-Kan El-page defined in 14.6, written [E'X1'. There

was already a map of vector spaces [EIX] -+ [E1 X] in V1 defined by

[E'X] -3 H=[ 1 Sqhj k = 1 H{- 1Sq k61, Z E [E1X],

205

and using the conjectural definitions of the operations on [E1 X], it is a map of chain com-

plexes. Indeed, we may calculate the differential in [E1X] exactly as we calculated in 14.6.

Thus, there is an induced map [E'X] -- [E2X] (where we write [EGX] for the homology of

the chain complex [E'X]). From now on, we will also assume:

Conjecture 2. The induced map [E2X] -+ [E 2X) is an isomorphism (of vector spaces).

This conjecture is not so unreasonable, since by Corollary 14.18 there is an isomorphism

of vector spaces [E2X] -+ [E2 X] given by mapping an element of [E'X] to the element of

[E2 X] of the same name, under the calculation of [E2X1 given by Corollary 14.18. In any

case, we assume no more than the stated conjectures.

16.2. The resulting differentials

We will now analyze the differentials dr applied to the various terms Sqj6vz. Define functions

f, n, e adm(A, T) - {0, 1,2,.. .}

which evaluate on a sequence I = (il, . . . , i1 ) as follows:

f(I) :=- 1; n(I) := ii + -- + ii; e(I) := il - i1_1- - - i = 2i, - n(I).

Define a function

a : adm(A, T) -+ Z

by a(I) := f(I) - 1 - (e(I) - T). Write 9 for the set consisting of those pairs of sequences

(J, I) appearing in the definition of [E'X], i.e.:

9 := {(J, I) 1I c adm(A, T), J C adm(A> i, f(I))}.

We may decompose 9 into three subsets:

' := f{(0, I) 1I adme(A, T)},

9' :={(J,I) E 9 \9 1 J = (J(), ... ,Ji), (J) = 0 or a() + 3 - ji < 0},

9" := {(J,I1)E 9 \9 I j = (jf(J),... ,j1), f(J) > 0 and a(I) + 3 - ji > 0}.

Every class 6'z for (0, I) E 9 e is a permanent cycle. On the other hand, we will prove:

Proposition 16.1. Assuming Conjectures 1 and 2, there is a bijective map g : 9' -s 9"

such that if g(J, I) = (J', I') then there is a differential dr : Sqhj{ - SqI{6, .

206

Using Conjecture 1 we may mimic the calculation of diSqj It made in 14.6. We find

that if I E adm(A, T) \ adme(A, T), SI2 survives to Ea(I)+l, at which point

a(I)+1 : 6)z Sq a(I)+26 - ,(16.1)

where we write I~ for the sequence (ii(l)-1,...,ii) obtained by removing the outermost

entry of I.

For an element J E adm(A>i, s) with J = (jj(J),.. . ,ji), and for any n > 2 - ji we will

write 1 ,J for the sequence

SJ :=- (j() + 2'(J)-n,. . ,j2 + 2n, ji + n) E adm(A>i, s + n).

There is a differential, obtained by applying SqJ to the da(I)+-differential (16.1):

d2ea(I)+: SqJ51i 2-+ Sq"a(I)JSq (l)+26,-z - Sq J+o-2,

where J+ := (Je(j) + 2f(J)~la(I), ... ,j2 + 2a(I), ji + a(I), a(I) + 2). We define the map g

by requiring that g(J, I) equals the pair (J+, 1-) whenever (J, I) E 9'.

Proof of Proposition 16.1. Firstly, we should check that g is well defined. Suppose first that

J= 0. Then we must have e(I) > T, so that

a(I) + 2 = f(I) - 1 - (e(I) - T) + 2 < f(I-) + 1,

a condition required for J+ to have any chance of lying in adm(A>i, (I-)). After this

initial check, it is easy to check the condition required of 7F(J+). Thus, (J, I) E 9 \ 9'. We

must also check that a(I) + 3 - (a(I) + 2) > 0, i.e. that a(I~) - a(I) > -1, which reduces

to the tautological condition e(I) e(I ). Thus g is well defined.

The injectivity of g is clear, but we must check its surjectivity. Suppose for this purpose

that (J, I) C 9", so that a(I) + 3 - ji > 0. We will begin by producing a differential

dj _1 : oJsz Sqh I~

with I+ a -admissible sequence (i7(i)+1 , ie(I), . , ii). For this, we need e(I+) e(I) (to

ensure admissibility of I+) and a(I+) = ji - 2, but we are otherwise unconstrained, as

ji 2, and the demand a(I+) > 0 will ensure that the additional 6(, is defined. Now

f(u+) h a (e(I+) - T) = ji - 2 -e(i+) - e( ) = a(I) + 3 - ji,

but we have assumed that a(I) + 3 - j, is non-negative, so we have no difficulty satisfying

207

the constraint a(I+) = i - 2.

Next we use the sequence _-a(I+)J-, where J- := (je(), . ,j2), to produce the required

differential

d2 -(J)-1a(I+)+1 S '-() Z+z -+ Sq: q I+z,

as long as either J- is empty or a(I+) + 3 - (j2 - a(I+)) < 0. If J- is non-empty, then the

second condition reduces to the condition that the concatenation of Da(I+)c1)-a(I+)J- with

the length one sequence (a(I+) + 2) is Sq-admissible, but this concatenation is J itself. l

Proposition 16.2. Assuming Conjectures 1 and 2, the differentials given in Proposition

16.1, along with those arising from them by taking products (without repetition) and applying

the Leibniz formula, are a complete set of differentials for the BKSS for this sphere.

Proof. Although the Er-page of the spectral sequence is not an exterior algebra for any

finite r, we are working in a spectral sequence of commutative F2-algebras. As such, the

differential is not sensitive to the difference between a polynomial algebra S(W)[x] and an

exterior algebra A (W)[x, x2 , X4 , X8 ... ] in which X 2 i is placed in the dimension of x2 . The

upshot is that the BKSS Er-page is isomorphic as a chain complex to an infinite coproduct

of exterior algebras, starting with

[E'] H L A(W)[Sqk 6 V Z].

(J,I)E9

That is, we may rely on the properties of the Steenrod operations to allow us to deal with

terms of the form x x x.

The differentials given in Proposition 16.1 are enough to eliminate all summands except

for

[Ec]j A(e) [f6vH Jk(0,I)E9e

which is isomorphic as an algebra to the target 7r,(Sfom).

Filtering the sets 9' and 9" by the length of the differentials associated with their ele-

ments, so that

9'. := {(J, I) E 9' if g(J, I) = (J', I') then n(J') + f(I') > n(J) + f(I) + r},

and 9'" := im(gl 9q), the proofs of Propositions 16.1 and 16.2 also prove:

Proposition 16.3. Assuming Conjectures 1 and 2, there is an isomorphism of chain com-

208

plexes, for r > 2:

[Er] H L A(W)[Sqk )J1Z] Lj L] A(e)[Sq'jkz)] L H A(%)LSqi L'(J,I)E-9 (JI)E-9r (JI)E9/

Moreover, the complete calculation of the BKSS for a finite connected model in s~om

now follows simply by taking the coproduct of non-unital differential graded algebras at each

page, with the appropriate grading shifts. For example:

[Er(Swom u Swo")] c [ES "O"] Li [ErSm].

209

210

Appendix A

Cohomology operations for Lie

algebras

In this appendix, we will prove that Priddy's definitions of cohomology operations for sim-

plicial (restricted) Lie algebras coincides with our own. There are three settings which we

are interested in: the categories sYie, s5ier and sLC(n) for n > 0. We will work in the third

setting in this appendix, as the proofs in the other two cases are strictly simpler.

A.1. The partially restricted universal enveloping algebra

For the following discussion, we will need one last category of graded vector spaces, Vn, an

object of which is simply the direct sum of an object V of V+ and a vector space 1,

V=V0 -- , 0 E ® ,.., E V.t>1 s,,,...,SJ>

Denote by A(n) the following category of graded augmented associative algebras. An object

of A(n) is a graded vector space A E V-Z such that A-'.,= F2 (1) is one-dimensional,

spanned by the unit of an associative unital pairing

A8n7,..., )1 A n,..., P1 - Sn+Pn,..,S1+P1

That is, A-' is not part of the data of A, but only a graded piece added to hold the unit.

Such an algebra is certainly augmented, and the augmentation ideal may be viewed as a

forgetful functor I : A(n) -- + (n), which sends A to the partially restricted Lie algebra

Att>_1 sn,...,s120

211

with bracket [x, y] := xy - yx, and restriction operation X[ 2] := X2 whenever x E A .

and not all of s,... , si zero.

The composite forgetful functor A(n) -- Z(n) -- V+ has a left adjoint, none other

than the free associative algebra functor FA(n) (also known as the tensor algebra functor).

The multiplicative unit 1 is placed in A- ', as is appropriate given the grading shift.

Moreover, the functor I has a left adjoint, U', the partially restricted universal enveloping

algebra functor, with U'L obtained as the quotient of FA(n)L by the two-sided ideal generated

by any [x, y] - xy - yx and by x[21 - X2 with x of restrictable degree. Indeed, there is a

composite of adjunctionsFZ(n) U

forget I

showing that U'o F ") '- FA(n). As in the non-restricted and fully restricted case, U'L is

naturally a Hopf algebra, having diagonal defined by the requirement Ax = 1 0 x + x 0 1

for x E L C U'L, and:

Lemma A.1 (PBW Theorem). If L E 1(n), then there is a natural increasing filtration

of U'L, the Lie filtration (by powers of (1) D im(L -> U'(L))), and the associated graded

algebra is naturally isomorphic to F2 [Lo] 0 E[LAo], where L = Lo G LAo is the decomposition

of L into the sum of its subspaces of in non-restrictable and restrictable degrees respectively.

Here, F2 [-] and E[-] denote the (shifted, unital) polynomial and exterior algebra func-

tors respectively, which differ from S(%) and A(%) only by the addition of the unit in

(F2 [-])-1..o and (E[-]) 10 . The unit 10 1 of this tensor product is in (F 2 [- E[-])-

as the product has a +1-shift in the cohomological dimension.

Lemma A.2. The prolonged functor U' : sL(n) -+ sA(n) preserves weak equivalences.

Proof. Suppose that L -- + L' is a weak equivalence in sZ(n). The Lie filtration makes

C*(U'L) -+ C*(U'L') a map of filtered commutative differential graded algebras, so there

is an induced map of the resulting spectral sequences. By Lemma A.1, the E0 -page of

the spectral sequence for U'L is the differential graded algebra C,(F2 [Lo] 0 E[Loo]). By

Dold's Theorem (2.4), the El-page is a functor (determined by the results of 5.4) of w,(Lo)

and 7r*(Lyo). As the induced maps 7r*(Lo) -- + 7r,(L') and 7r*(Lgo) --+ ir(L'o) are

isomorphisms, the map of spectral sequences is an isomorphism from El.

A.2. The proof of Proposition 6.12

In this section we will demonstrate Proposition A.3, which is stated for partially restricted

Lie algebras L E sZ (n), but can be reinterpreted for objects of syie or syier as necessary.

From this result, Propositions 6.12 and 8.9 follow.

212

Let L C s12(n) be almost free on a fixed choice of subspaces V 9 Lp. We will use a

bisimplicial model for WU'L:

Bpq := qU'Lp = (U'L )®q E ssV,

which in each simplicial level p is the standard simplicial bar construction for calculation of

TorU'LP (F2 , F2 ) (c.f. [47, 1]). There are natural equivalences

C|BI -- Tot(C*C*B) = Bar(C*U'L) ~ C*WU'L,

so that 7r*DIBI ~ HAL. Here, we have written Bar for the bar construction of [28, 7], and

the final equivalence is the homomorphism of [28, Theorem 20.1]. What is a little less well

known is that there is a natural weak equivalence of simplicial coalgebras underlying this

equivalence of chain complexes, given in [15, Theorem 1.1]. A simple construction of such a

map IBI -- + WU'L is, in simplicial level n:

do 2 0 - -9 .. do":('X)4- U'Xn_ -I . U'X0,i

where we use the conventions of [43, 5] to define W.

As such, the operations defined by Priddy on H, correspond, under this equivalence,

to those that we define on 7r*DIBI by the formulae

Sq (7 n DIB 2 7r Dn+k DS2 |BI 7rn+k DIBI);

yt : (S2(7r*DIBI) -47r*S2DJBJ -+ g7r*DS2JBI - : r*DIBI).

where AB is the bisimplicial cocommutative coalgebra diagonal:

AB : (f(U'L) -A) B(U'L 0 U'L) ~ B(U'L) 0 B(U'L)).

Thus, we may forget the functor V, and restrict our attention to the object B with this

coalgebra map. We are also going to use the simplicial chain complex Q C s ch+Vn

QZMn)L., if * = 1;

Q. : F 2 {1}, if * = 0;

0, otherwise.

with zero differentials in each simplicial level. Of course, we mean that 1 E (Q,000 1O'

There is a map of simplicial chain complexes r : N*B. -+ Q*,, defined in level p by the

213

identification NO'Bp = F2 {1} = Qpo and the composite:

NrvBp = IU'Lp -- * IU'Lp/(IU'L,)2 ~ pn) Lp.

Proposition A.3. The composite

N* BI ~ Tot(NhN/'B) - Tot(NQ.) = F2 D EN*Q4(n)L

is a weak equivalence of chain complexes under which the operations on 7r*D|B| defined

using AB correspond to the operations Sqk o qexk-1 and p:= 0 [ext on

7r*(D(QfZ(n)L)) =: H (n)L.

We will prove this proposition using the external spectral sequence operations of 13.1

in the spectral sequence of B. By E2 , the only interesting non-zero entries of this spectral

sequence lie on the horizontal line q = 1, so that Singer's operations will prove very un-

interesting without modification. Our method will be to perform such a modification by

using the chain homotopy h (defined shortly) to shift the horizontal operations one higher

in filtration. The shifted homotopy operations will preserve the line q = 1, and will abut to

operations on E.. that satisfy the same relations as those on IBL. As the abutment filtration

is trivial, they must satisfy the same relations at E2 . Finally, we will note that what we

have produced at E 2 is the definition of the Steenrod operations from 6.8.

As L is levelwise free, the evident map FA(n)V, -+ U'L, is an isomorphism for each

p, and we define a vertical homotopy h : NvBp -- + N+1 Bp by the following formulae (in

which the vij are taken to be in V C Lp C U'LC):

hq : NvBU' L - Nv*1A'L

[Vil I .. Vik- 1 VikVik+l I> [Vi lI.. Al IVik~l I

length 1 bars [vil .I i, 0.

This homotopy is of the same type as that used in 8, 9 and [47, Proof of Theorem 5.3], and

commutes with all of the horizontal simplicial structure except do, so that dhhq + hqd

dbhq + hqdh.

Lemma A.4. Under the map (Id + hq--1dv + dvhq) : NvfBU'Lp -+ NqvfBU'Lp,

[rq, - - - - -. I-| - I, 0 unless r = 1 in which -nse

[vil] [Vi l.

214

Lemma A.5. The composite

N NvB ' N Nv(B B) DY A h(NB NvB) N (Ql 9 Qj)

vanishes except on terms [x~y] with x and y generators of Lp, which have image x 0 y.

Proof. A generic element of the domain is a sum of terms [xi ... xIyli ... yj], with xi, ... , xj

and yi,... , yj in V C Lp. This element maps under AB to the following sum, taken over

all sequences of exponents a, ... ,ajbl,.. . , bj E {0, 1}:

[ai ... x- b4 ... Ybj]( [X-ai .-.-. X y-a -b1 ... Y1-bj] C NhNv(B ® B),

and (Do)* annihilates all terms except for those in which all ai are 1 and all bj are 0, leaving

[x1.x1] ® [y1-.yi] C N (NjB 9 N'B).

Finally, r O r annihilates this term unless I = J 1. 0

Lemma A.6. The composite

NP+ 1NrB A 4 NP+ 1 Nj'(B ® B) N+N +1(Nj B 9 Nv B) -*4 No+1 (Qe. Q91)

vanishes except on terms [xy] with x and y generators of Lp+i, which have image xoy+yox.

Proof. A generic element of the domain is a sum of terms [xi .. x-], with x1,... , xj in

Vp+l C Lp+. This element maps under AB to

al ... xaj (g [xI-al .. X4-ai]

As {Dk} was chosen to be a special k-cup product, (D')* acts as the identity in this case.

Finally, r 9 r annihilates this term unless I = 2 and a, 4 a2.

Lemma A.7. There is a commuting diagram:

d hfl+k+hn+k-143 k N+k-(N[B 0 N'B)n~~~~~k~~ r~- nk (vr2)NB

Nh +QZ (n) L -Nflh Y n L- L) - Nh~k (QL(n)L 0& QZ(fl)L)na+k QM)Lnb+k-1Q L* LvL >na+k-1

Proof. Write LHS = (r 0 r) o (Do)* o AB o (d h + hc4) and RHS = 0(n) 0 r. Consider first

an element e = [viv2 .. b] of N~h NjB with b > 2. By definition, r vanishes on such an

element, so that RHS(e) = 0. Lemma A.5 states that the map (r 0 r) o (Do)* 0 AB vanishes

215

except on expressions of the form [uIw] for u, w C Vn+k-1. However, the expressions of

this form appearing in dhh(e) coincide with such expressions in hdh(e), so that there is a

cancellation, and LHS(e) = 0 as hoped.

Next, consider an element [v] of Nh+kNIB. As h[v] = 0, and in light of Lemma A.5,

LHS([v]) equals the quadratic part of AV, after writing dhv as an expression in elements of

Vn+k-1. This is exactly the description given in Lemma 6.3 of RHS([v]) = O)(n)(v). E

Proof of Proposition A.S. Fix a cocycle a E D(NnQ'(n)L). Then a may be viewed an a

permanent cocycle in [ZD(Q,*)]~"> in the spectral sequence obtained by dualizing Q..

Singer [53, (2.14)] defines an operator Sk on the total cochain complex of a bisimplicial

coalgebra which induces the cohomology operation Sqe. We will apply the chain-level

operator Sk to the class r*a C [ZOB]n>1 . As a is a permanent cycle, d(r*a) = 0, and

Singer's expression simplifies to:

Sk(r*a) := A*Kn+l-k(r*a 0 r*a) = Ti + T2 , where:

T :A*DoDn+1 -kq(r*a o9 r*a) c D(Nh+k- N'B)

T2 :=A* D(TD j-k T)(r*a 0 r* a) c D(N hk NvB).

Our method will be to compress each of these terms into filtration one higher, using the

cochain homotopy h* : D(NhNvB) -- + D(NhN_11B). Using Lemma A.4:

(Id + dvh* + h*dv)T1 = 0 and (Id + dvh* + h*d)T2 = 0.

The first equation holds as (Id + hdV + dvh) is zero on NIB. For the second equation, on

NT B, (Id + hdv + dvh) is the projection onto terms of the form [v], yet Lemma A.6 shows

that the composite

((r 9 r) o (T(Dnk)*T) o (D')* 0 AB) : N h kNB -- + N,(+k(Q. 1 0 Q.1)

vanishes except on terms of the form [vw] (recall that r commutes with the horizontal

simplicial structure).

As dhh* + h*dh increases filtration, we have compressed Sk(r*a) to the filtration n + k

expression (dhh* + h*dh)T, modulo even higher filtration. The commuting diagram of

Lemma A.7 is the left square in a larger commuting diagram:

(nO *A o (dh 1 _,h) (n~n+1-k)*Nh ' \ "> - B 9 NB)\h NNjB O NhNvBn+kNi) 'Vnf+k (N3,N+B-

kQ r O (n k O ) (D4nrlr ) kL I r )

NPhkQ"~)L -~~--N 7h n)~() 0 ~~L) h NhQL(n)L ®9 NhQC )L

216

Now D(NhQ47')L 0 N,,Q4()L) contains the cocycle (a o0 a), and pulling 0(a 0 a) back

to D(Nh+kNB) along the lower composite yields r*,O* (SqkJ(a). Pulling back along the

upper composite yields the E2 representative of the shifted version of Singer's operations.

Both spectral sequences collapse at E2 and induce trivial filtrations on their shared target, so

that understanding the shifted operations at E2 is equivalent to understanding the operations

on r*D IBI, which we do: they equal Priddy's operations on 7r*WU'L [48, 5]. As r* is an E2 -

equivalence, this proves the result. A simple modification proves the result for pairings. E

A.3. The Chevalley-Eilenberg-May complex

Suppose that M C (n) is a partially restricted Lie algebra of finite type (not simplicial).

One can define a differential coalgebra, the Chevalley-Eilenberg-May complex, to be the

subcoalgebra X'(M) := E[Mo] 0 F[Mo] of the divided power Hopf algebra F[M] with its

usual coalgebra structure (c.f. [40, p. 141]), graded as follows. The Hopf algebra F[M] is to

be V 1 -graded, with product and divided square operations

0MF [M ] , -+ F [M t+t + 1,s+A' ,. .

-Y2 : F[M] +,8 . -4 F[AI]+1, 2Sn,...,2s'

generated by the subspace

r,[M] -= Ms,...,lI

and we define X'(M) to be the coaugmentation coideal of the subcoalgebra spanned by

those expressions

yri(yi)- - rm(ym) (with yi, ... ,ym E M homogeneous)

for which ri < 1 when yi E Mo (i.e. y, E M . The coalgebra structure map and

differential are the restriction to E[Mo]or[Myo] of those given in [40, p. 1411 (after tensoring

the formula [40, (6.19)] down to a formula on X(M), which kills the first term E liyi).

This differs by a shift from the standard definitions, given in [16] in the unrestricted

setting, and given in [40] in the restricted setting. It also differs from those definitions in

that we have taken the coaugmentation coideal. Correspondingly, X'(M) is a shift of the

homology (in the sense of [47]) of the associated graded algebra appearing in the partially

restricted PBW Theorem, Lemma A.1.

Now let L - B4 c)M E sC(n). Using the equation |B| = B.(U'M) of simplicial coal-

gebras and May's injection [40, Theorem 18 and (7.8)] of X'(M) into the bar construction,

217

there are maps:

EX'(M) -- + N.B.(U'M) ~Tot(N)hN/B) - Tot (Ng Q.o*) = IF2{1} (D EN*Q4(")L.

Now the first map, after the suspension E shift homological degree, is a degree preserving

map of differential coalgebras, both of which only have a shift in the cohomological degree

t. In light of this discussion, Proposition A.3 implies Proposition A.8 (in which we move

back into the notation S(W) and A(%) for the non-unital commutative and exterior algebra

monads).

Proposition A.8. H$,) M may be calculated, as a non-unital commutative algebra, as

the cohomology algebra of the differential graded algebra D(X'(M)), where D(X'(M)) is the

non-unital commutative algebra D(X'(M)) = A(%)[DMo] Li S(W)[DMAo]. This algebra is

vn+l-graded, generated by its subspace

(D(X'(M)))t'"'.'i = (DM)t'.

and has grading shifted product

D (-V'(M )) ' * "''' 0 D ( '(M )) ,' ' I -+.D(.'(M )) +t+1 1

Recall that the coproduct A Li B of non-unital commutative algebras is the direct sum

A E (A ® B) e B.

We are particularly interested in the case that M is trivial as a Lie algebra, but may still

have non-zero restriction. In this case, the restriction is in fact a linear map, and we may

write V/ : DM -+ DM for its dual (a map which we consider to be everywhere defined,

but necessarily equal to zero on Mo). Examination of [40, (6.19)] shows:

Proposition A.9. If M has bracket zero, then the differential on the cohomological dif-

ferential graded algebra D(X'(M)) is defined on generators a ( DM C D(X'(M)) by the

formula

a ,( ) 2.

218

Bibliography

[1] Michel Andre. Methode simpliciale en algebre homologique et alg~bre commutative. Lec-

ture Notes in Mathematics, Vol. 32. Springer-Verlag, Berlin-New York, 1967, pp. iii+122.

[2] Jon Beck. "Distributive laws". In: Sem. on Triples and Categorical Homology Theory

(ETH, Ziirich, 1966/67). Springer, Berlin, 1969, pp. 119-140.

[3] David Blanc and Christopher Stover. "A generalized Grothendieck spectral sequence".

In: Adams Memorial Symposium on Algebraic Topology, 1 (Manchester, 1990). Vol. 175.

London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press, 1992, pp. 145-

161.

[4] A. J. Blumberg and E. Riehl. "Homotopical resolutions associated to deformable ad-

junctions". In: ArXiv e-prints (Aug. 2012).

[5] A. K. Bousfield. "A vanishing theorem for the unstable Adams spectral sequence". In:

Topology 9 (1970), pp. 337-344.

[6] A. K. Bousfield. Homogeneous functors and their derived functors. 1967.

[7] A. K. Bousfield. "On the homology spectral sequence of a cosimplicial space". In: Amer.

J. Math. 109.2 (1987), pp. 361-394.

[8] A. K. Bousfield. Operations on derived functors of non-additive functors. 1967.

[9] A. K. Bousfield and D. M. Kan. "A second quadrant homotopy spectral sequence". In:

Trans. Amer. Math. Soc. 177 (1973), pp. 305-318.

[10] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations. Lec-

ture Notes in Mathematics, Vol. 304. Berlin: Springer-Verlag, 1972, pp. v+348.

[11] A. K. Bousfield and D. M. Kan. "Pairings and products in the homotopy spectral

sequence". In: Trans. Amer. Math. Soc. 177 (1973), pp. 319-343.

[12] A. K. Bousfield and D. M. Kan. "The homotopy spectral sequence of a space with

coefficients in a ring." In: Topology 11 (1972), pp. 79-106.

[13] A. K. Bousfield et al. "The mod - p lower central series and the Adams spectral

sequence". In: Topology 5 (1966), pp. 331-342.

219

[14] H. Cartan. "Puissances divis6es". fre. In: Seminaire Henri Cartan 7 no. 1 (1954-1955),

pp. 1-12.

[15] A. M. Cegarra and Josu6 Remedios. "The relationship between the diagonal and the

bar constructions on a bisimplicial set". In: Topology Appl. 153.1 (2005), pp. 21-51.

[161 Claude Chevalley and Samuel Eilenberg. "Cohomology theory of Lie groups and Lie

algebras". In: Trans. Amer. Math. Soc. 63 (1948), pp. 85-124.

[17] M. Ching. "Homotopy Operations on Simplicial Algebras over an Operad". Available

at http: //www3.amherst. edu/~mching/Work/homotopy-operat ions. pdf.

[18] M. Ching and J. E. Harper. "Derived Koszul duality and TQ-homology completion of

structured ring spectra". In: ArXiv e-prints (Feb. 2015).

[19] E. B. Curtis. "Lower central series of semi-simplicial complexes". In: Topology 2 (1963),

pp. 159-171.

[20] E. B. Curtis. "Simplicial homotopy theory". In: Advances in Math. 6 (1971), 107-209

(1971).

[21] loannis Dokas. "Quillen-Barr-Beck (co-) homology for restricted Lie algebras". In: J.

Pure Appl. Algebra 186.1 (2004), pp. 33-42.

[22] Albrecht Dold. "Homology of symmetric products and other functors of complexes".

In: Ann. of Math. (2) 68 (1958), pp. 54-80.

[23] Albrecht Dold. "Uber die Steenrodschen Kohomologieoperationen". In: Ann. of Math.

(2) 73 (1961), pp. 258-294.

[24] Albrecht Dold and Dieter Puppe. "Homologie nicht-additiver Funktoren. Anwendun-

gen". In: Ann. Inst. Fourier Grenoble 11 (1961), pp. 201-312.

[25] W. G. Dwyer. "Higher divided squares in second-quadrant spectral sequences". In:

Trans. Amer. Math. Soc. 260.2 (1980), pp. 437-447.

[26] W. G. Dwyer. "Homotopy operations for simplicial commutative algebras". In: Trans.

Amer. Math. Soc. 260.2 (1980), pp. 421-435.

[27] W. G. Dwyer, D. M. Kan, and C. R. Stover. "An E2 model category structure for

pointed simplicial spaces". In: J. Pure Appl. Algebra 90.2 (1993), pp. 137-152.

[281 Samuel Eilenberg and Saunders Mac Lane. "On the groups of H(1I, n). I". In: Ann. of

Math. (2) 58 (1953), pp. 55-106.

[29] Samuel Eilenberg and John C. Moore. "Limits and spectral sequences". In: Topology

1 (1962), pp. 1-23.

220

[30] B. Fresse. "Koszul duality of operads and homology of partition posets". In: ArXiv

Mathematics e-prints (Jan. 2003).

[31] B. Fresse. "On the homotopy of simplicial algebras over an operad". In: Trans. Amer.

Math. Soc. 352.9 (2000), pp. 4113-4141.

[32] Paul G. Goerss. "A Hilton-Milnor theorem for categories of simplicial algebras". In:

Amer. J. Math. 111.6 (1989), pp. 927-971.

[33] Paul G. Goerss. "On the Andr6-Quillen cohomology of commutative F2-algebras". In:

Astdrisque 186 (1990), p. 169.

[34] Paul G. Goerss and John F. Jardine. Simplicial homotopy theory. Vol. 174. Progress

in Mathematics. Basel: Birkhduser Verlag, 1999, pp. xvi+510.

[35] Thomas G. Goodwillie. "Calculus. II. Analytic functors". In: K-Theory 5.4 (1991/92),

pp. 295-332.

[36] Philip Hackney. "Homology operations and cosimplicial iterated loop spaces". In: Ho-

mology Homotopy Appl. 16.1 (2014), pp. 1-25.

[37] Philip Hackney. "Operations in the homology spectral sequence of a cosimplicial infi-

nite loop space". In: J. Pure Appl. Algebra 217.7 (2013), pp. 1350-1377.

[38] K. Hess. "A general framework for homotopic descent and codescent". In: ArXiv e-

prints (Jan. 2010).

[39] Mark Hovey, John H. Palmieri, and Neil P. Strickland. "Axiomatic stable homotopy

theory". In: Mem. Amer. Math. Soc. 128.610 (1997), pp. x+114.

[40] J. P. May. "The cohomology of restricted Lie algebras and of Hopf algebras". In: J.

Algebra 3 (1966), pp. 123-146.

[41] J. Peter May. Simplicial objects in algebraic topology. Van Nostrand Mathematical

Studies, No. 11. D. Van Nostrand Co., 1967, pp. vi+161.

[42] John McCleary. A user's guide to spectral sequences. Second. Vol. 58. Cambridge

Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2001,

pp. xvi+561.

[43] Haynes Miller. "The Sullivan conjecture on maps from classifying spaces". In: Ann. of

Math. (2) 120.1 (1984), pp. 39-87.

[44] Brian Munson and Ismar Voli6. "Cubical Homotopy Theory". In preparation to be

published by Cambridge University Press. 2014.

[45] Joseph Neisendorfer. Algebraic methods in unstable homotopy theory. Vol. 12. New

Mathematical Monographs. Cambridge: Cambridge University Press, 2010, pp. xx+554.

221

[46] L. Pereira. "A general context for Goodwillie Calculus". In: ArXiv e-prints (Jan. 2013).

[47] Stewart B. Priddy. "Koszul resolutions". In: Trans. Amer. Math. Soc. 152 (1970),

pp. 39-60.

[48] Stewart B. Priddy. "Primary cohomology operations for simplicial Lie algebras". In:

Illinois J. Math. 14 (1970), pp. 585-612.

[49] Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Berlin:

Springer-Verlag, 1967.

[50] A. Radulescu-Banu. "Cofibrance and Completion". PhD thesis. MIT, Dec. 2006.

[51] James W. Schlesinger. "The semi-simplicial free Lie ring". In: Trans. Amer. Math. Soc.

122 (1966), pp. 436-442.

[52] Lionel Schwartz. Unstable modules over the Steenrod algebra and Sullivan's fixed point

set conjecture. Chicago Lectures in Mathematics. University of Chicago Press, Chicago,

IL, 1994, pp. x+229.

[53] William M. Singer. Steenrod squares in spectral sequences. Vol. 129. Mathematical Sur-

veys and Monographs. American Mathematical Society, Providence, RI, 2006, pp. xii+155.

[54] Dev P. Sinha. "The topology of spaces of knots: cosimplicial models". In: Amer. J.

Math. 131.4 (2009), pp. 945-980.

[55] Christopher R. Stover. "A van Kampen spectral sequence for higher homotopy groups".

In: Topology 29.1 (1990), pp. 9-26.

[56] James M. Turner. "Operations and spectral sequences. I". In: Trans. Amer. Math. Soc.

350.9 (1998), pp. 3815-3835.

[57] J. H. C. Whitehead. "On adding relations to homotopy groups". In: Ann. of Math. (2)

42 (1941), pp. 409-428.

222

archnes - DSpace@MIT

Documents