The Kahn-Priddy Theorem202014/JensJako… · Abstract In this thesis we will prove The Kahn-Priddy Theorem in stable homo-topy theory. The theorem states that there exists a morphism

U N I V E R S I T Y O F C O P E N H A G E N

Jens Jakob Kjær

The Kahn-Priddy Theorem

Thesis for the Master degree in Mathematics

Department of Mathematical Sciences

University of Copenhagen

Advisor Markus Szymik

April 11th, 2014

Abstract

In this thesis we will prove The Kahn-Priddy Theorem in stable homo-topy theory. The theorem states that there exists a morphism of spectraRP∞ → S0, which is surjective on the positive 2-localized stable homotopygroups. We will in fact prove a stronger statement: Jones’ strengtheningof The Kahn-Priddy Theorem. To this end we will recall a number of wellknown results from stable homotopy theory and discuss the construction ofthe stunted real projective spectra. After proving our main theorem we willturn our attention to some filtration and to certain elements of the stablehomotopy groups of the spheres, which relate to the stunted real projectivespectra. The spectral sequence, which naturally arises from the stunted realprojective spectra will be the next object of our attention. We will provethat the sequence converges to π∗S

−12

and we will see, what Jones’ Kahn-Priddy Theorem reveals about the situation. We will also calculate a fewdifferentials and make a note of the periodicity. Lastly we will see how muchof the above which translate to the stunted complex projective spectra.

Resume

I det foreliggende speciale vil vi bevise Kahn-Pridy-Sætningen i stabil ho-motopi teori. Sætningen siger, at der findes en morfi af spektra RP∞ → S0

som er surjektiv pa de positive 2-lokaliserede stabile homotopi grupper. Vivil bevise en stærkere udgave af resultatet i form af Jones’ Kahn-Priddy-sætning. For at gøre dette vil vi genkalde os en del velkendte resultater frastabil homotopi teori samt diskutere konstruktionen af afskarne reelle pro-jektive spektra. Efter, at vi har vist vores hovedresultat, vil vi vende voresopmærksomhed mod en vis filtering samt mod visse elementer af de stabilehomotopi grupper af sfæren, som releterer til de afskarne reelle projektivespektra. Det næste vi vil studere er den spektral følge, som naturligt opstarfra de afskarne reelle projektive spektra. Vi vil bevise den konvergerer modS−1

2og studere, hvad Jones’ Kahn-Priddy-sætning fortæller os om situatio-

nen. Vi vil ogsa beregne nogen differentialer og notere os periodiciteten. Tilsidst vil vi undersøge, hvor meget af det foregaende, som kan oversættes tilsituationen med afskarne komplekse projektive spektra.

Introduction

One can take the study of algebraic topology to be the study of calculat-ing algebraic invariants of topological spaces and we have many differentinvariants to choose from. The most interesting of these are the homotopygroups as it is well known that they detect homotopy equivalences of CW-complexes. This property alas comes with a cost as they are near impossibleto calculate even in simple examples and therefore seem to be reserved for arole more theoretical than computational. There are however ways to easethe calculations albeit with some acceptable loss of information.

In 1937 Hans Freudenthal proved his now famous suspension theorem,stating that πi(X) ∼= πi+1(ΣX) if the connectivity of X is high compared toi, where Σ denotes the reduced suspension. Since suspension increases theconnectivity of a space it the holds that

πi(X)→ πi+1(ΣX)→ πi+2(Σ2X)→ . . .

must stabilize at some point for any space X. We call the colimit of thissequence the i’th stable homotopy group of X. It turns out that there arevarious methods for computing these groups, e.g., the classical Adams Spec-tral Sequence developed by Frank J. Adams in 1958, and they are thereforeof great interest. This interest further developed into the field of stable ho-motopy theory, wherein we study up to homotopy the maps which survivethe suspensions. It turned out that a generalization of spaces was needed,leading to the notion of spectra, which generalize the notion of a topologicalspace X and its suspensions. Further, many great classical results and con-structions have analogues pertaining to spectra and stable homotopy theory.Of special interest among spectra is the sphere spectrum which fulfils thesame roles as do the the spheres in our normal homotopy theory.

A different approach to computing homotopy groups (or as we will see inthis thesis stable homotopy groups) is by means of localizations. The conceptis that we want a homotopy theory which only detects certain structures.For the thesis at hand we will mostly concern ourselves with localizationsof spectra in such a way that we only see the 2-torsion part of the stablehomotopy groups, as this is a well studied case.

The approach we will follow in this thesis is to reprove Jones’ strength-ening of The Kahn-Priddy Theorem.

iii

INTRODUCTION

The Kahn-Priddy Theorem states that there exists a map of suspensionspectra

τ ′ : RP∞ → S0

which is surjective on positive stable 2-localized homotopy groups. The the-orem was first stated in [KP72]. Later Jones discovered a natural strength-ening of the statement [Jon85]. For all k ∈ N0, Pk denote the stuntedreal projective space, RP∞/RP k−1, then there is a generalization of thesespaces to the stunted projective spectra P−s existing for all s ∈ Z. Jones’strengthening states that there are morphisms of spectra

ι : S−1 → P−s

which are trivial on the (t − 1)’th stable homotopy group whenevers ≤ t. This result then implies The Kahn-Priddy Theorem since we havethe following cofiber sequence

S−1 ι // P−1// P0

τ // S0

and ”restricting” τ to P1 = RP∞ gives us τ ′. Jones’ argument was latersimplified by Miller [Mil90].

Miller also indicated that the results can be best understood in terms ofMahowald’s stable EHP spectral sequence, filtration and root invariants.

The twofold aim of this thesis is therefore to

1. give an in-depth proof of Jones’ theorem based on Miller’s paper, and

2. give a detailed presentation (with some illustrating calculations) of thepart of the spectral sequence that this illuminates.

A break down of the chapters is as follows: In Chapter 1 we will give thepreliminaries required for the rest of the thesis, recalling many definitionsand theorems from stable homotopy theory. Chapter 2 will concern itselfwith the construction of the stunted real projective spectra P−s. In Chapter3 we will state and prove Jones’ strengthening of The Kahn-Priddy Theoremand spend some time on its effects on the Mahowald filtration and rootinvariants. In chapter 4 we will construct the stable EHP spectral sequencerelating to the tower of stunted projective spectra and do some calculationsof the differentials in it. Chapter 5 we will turn our attention to the stuntedcomplex projective spectra and the complex Mahowald filtration and rootinvariants, as well as constructing a spectral sequence very similar to the onefor real projective spectra. Appendix A, appendix B, and appendix C willcontain Bruner’s calculations of the E2-terms of the classical Adams SpectralSequence relating to the real respectively complex stunted projective spectraand the sphere spectrum and the mod-2 Moore spectrum.

iv

INTRODUCTION

Acknowledgements

I would like to take this opportunity to thank several people who have beenof considerable assistance to me in the work of writing this thesis. First andforemost of course my advisor Professor Markus Szymik, who has not onlyput up with my many questions but conveyed to me, his great fascination,with stable homotopy theory. Also I would like to thank Dr. Irakli Patchko-ria for his open door policy, which has considerably aided me, giving me asecond place to come with my many questions.

I further extend my thanks to Kristian Kjær and Amalie Høgenhavenwho have diligently proofread the thesis, resulting in significant improve-ments.

Lastly I would like to thank my family and friends for putting up withme while I have been working on this thesis.

v

vi

Contents

Introduction iii

1 Preliminaries 11.1 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Stable Homotopy Theory . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Homology and cohomology theories . . . . . . . . . . . 91.3 Homotopy limits and colimits . . . . . . . . . . . . . . . . . . 111.4 Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 The Adams Spectral Sequence . . . . . . . . . . . . . . . . . 131.6 S-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Geometry 162.1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Thom Spaces . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Quadratic Construction . . . . . . . . . . . . . . . . . 19

2.2 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Thom Spectra . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Quadratic Construction . . . . . . . . . . . . . . . . . 232.2.3 The Spectra Pn . . . . . . . . . . . . . . . . . . . . . . 232.2.4 The CW-structure . . . . . . . . . . . . . . . . . . . . 24

2.3 Atiyah Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 clASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Jones’ Kahn-Priddy Theorem 283.1 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Mahowald filtration and root invariants . . . . . . . . . . . . 32

4 Spectral Sequence 374.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 The E1 page . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Jones’ Kahn-Priddy Theorem and the E∞-page . . . . . . . . 434.5 Localizations of the spectral sequence . . . . . . . . . . . . . 444.6 The Root Invariant . . . . . . . . . . . . . . . . . . . . . . . . 45

vii

CONTENTS

4.7 Some of the d2’s in the spectral sequence . . . . . . . . . . . . 46

5 The Complex Projectives 545.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.1 clASS . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 The Complex Mahowald Filtration and Root Invariant . . . . 575.3 The Spectral Sequence . . . . . . . . . . . . . . . . . . . . . . 59

5.3.1 The set up . . . . . . . . . . . . . . . . . . . . . . . . 595.3.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . 60

Outlook 64

A E2 for clASS of P−s 67

B E2 for clASS of CP−s 77

C E2 for clASS of S0 and M(2) 87

Bibliography 91

viii

Chapter 1

Preliminaries

In this chapter we will list a number of definitions and results important forthe thesis. This is in no way meant to be an in-depth introduction to stablehomotopy theory. Rather this should be taken as a brief reminder of sometopics familiar to readers who already have had an introduction to stablehomotopy theory and category theory. The preliminaries are taken from anumber of sources but a more thorough treatment can be found in [Swi02]and [Rav92] among many other places.

Stable homotopy theory started with the result of Freudenthal on the sus-pension functor: For a n-connected space X then πkX ∼= πk+1ΣX wheneverk ≤ 2n. As suspension increases connectivity the colimit over this processmust hence be obtained at some point. These groups were called the stablehomotopy groups of X. It quickly became apparent that many results andnotions from algebraic topology had an equivalent stable version. It alsocame to light that if this theory was to be fully developed more ”spaces” wasneed and hence the more abstract notion of a spectrum emerged.

1.1 Spectra

We will start by defining our basis objects, spectra. These serve as gener-alizations of the concept of a space and all of its suspensions and as suchmuch of our intuition carries over. But before we can do this we need toremind ourselves of some elementary constructions of topology, namely thesuspension and loop space for a topological space.

Definition 1.1.1. Given a pointed topological space (X,x0) we take theunreduced suspension to be the pointed space of SX := X×I/ ∼, with ∼ theequivalence relation induced by (x, t) ∼ (x′, t′) if (x, t) = (x′, t′), t = t′ = 1or t = t′ = 0. Then we define the (reduced) suspension ΣX := SX/x0×I,with [x0×I] as the base point. Given a continuous pointed map f : X → Ywe see that f × IdI : X × I → Y × I induces a continuous pointed mapΣX → ΣY . We call this map Σf .

1

Section 1.1 Spectra

We write ΣnX for Σ(Σn−1X) with the convention that Σ0X = X. IfX is a CW-complex then so is ΣX. Further we know the cell structure ofthe complex ΣX since for each non base point i-cell of X, eiα, then ΣXhas a cell ei+1

α = Σeiα. Furthermore taking the suspension of a space raisesits connectivity by one. Lastly we recall the connection between takingsuspension and the smash product since it is well known that ΣX = S1 ∧Xfor any pointed space X and ΣnX = Sn ∧X.

Definition 1.1.2. Given a pointed topological space (X,x0) we define theloop space of X, ΩX, to be the space of pointed maps from S1 to X giventhe compact open topology and basepoint the constant loop in x0.

We recall that Σ and Ω are adjoint functors, and that ΩnX = ΩΩn−1X(with Ω0X = X) can be thought of as the space of pointed maps from Sn

to X with the same topology and basepoint as above, and this is the adjointfunctor to Σn.

We are now ready to define what we will mean by a spectrum.

Definition 1.1.3.

• A spectrum X is a collection of pointed topological spaces Xn forn ∈ N0 together with structure maps σn : ΣXn → Xn+1, where eachσn is a pointed continuous map.

• Given spectra X and Y we say that a strict map between spectraf : X → Y is a collection of continuous pointed maps fn : Xn → Ynfor all n such that

ΣXnΣfn //

σXn

ΣYn

σYn

Xn+1fn+1 // Yn+1

commutes for all n, where σX and σY are the structure maps for Xand Y respectively.

Instead of giving the structure maps as σ : ΣXn → Xn+1 we can alsospecify ω : Xn → ΩXn+1, since Σ and Ω are adjoint functors. We note thatthere are several ways of equipping a spectrum with additional structure, butdue to lack of time we will not go into that in this thesis. The most intuitiveclass of spectra is that of suspension spectra, which will be important as itformalizes how spectra generalize spaces.

Definition 1.1.4. Given a pointed space X we can define the suspensionspectra of X, Σ∞X, to be the spectra (Σ∞X)n = ΣnX, with structure mapsthe identities.

2

CHAPTER 1 PRELIMINARIES

In most cases we will not be bothered to make the distinction between apointed space and its suspension spectrum, since the context will make thedistinction clear. Since this entire topic grew out of the suspension functorfor spaces we will need something similar for spectra.

Definition 1.1.5. Given a spectrum X we can define the k-fold suspensionΣkX to be the spectrum given by (ΣkX)n = Xn+k with the obvious structuremaps and the convention that Xi = ∗ if i < 0. If k < 0 we will also call itthe (−k)-fold desuspension.

A spectrum essential to our work will be the sphere spectrum which willfulfill the role of the spheres in standard homotopy theory.

Definition 1.1.6. For all n ∈ Z, Sn denote the spectrum ΣnΣ∞S0. If n = 0we call it the sphere spectrum.

Note, we allow for n < 0 as opposed to regular spaces.

In unstable homotopy theory CW-complexes are of paramount impor-tance and for the benefit of the additional theorems which holds for CW-complexes one often restricts ones attention to the case of CW-complexes,away from the general notion of topological spaces. Therefore we will nowgeneralize the notion of CW-complexes to the notion of CW-spectra.

Definition 1.1.7.

• A CW-spectrum is a spectrum where all the Xn’s are pointed CW-complexes and the structure maps are inclusions of pointed subcom-plexes.

• The (stable) cells of a CW-spectra X, is the cells, different from thebase point cell, of the spaces Xn under the equivalence relation thateiα ∈ Xn is the same as its image under the structure maps, i.e.,ei+1α ∈ Xn+1. Thus the cells of X consist of cells ek+n

α ∈ Xn for alln > nα for some nα ∈ N. We say that such a cell is of dimension k.

• A collection of subcomplexes X ′n ⊂ Xn, such that σn(ΣX ′n) ⊂ X ′n+1 asa subcomplex, is called a subspectrum.

• A subspectrum is called cofinal if for any eiα that is a stable cell of Xn

there is some k ∈ N such that the cell Σkeiα is a cell of X ′n+k.

• Given X,Y CW-spectra a map of CW-spectra f : X → Y is a strictmap of spectra f ′ : X ′ → Y for some cofinal subspectra X ′ of X.

• A CW-spectrum is said to be finite if it has only finitely many cells.It is said to be of finite type if it only has finitely many cells of eachdimension.

3

Section 1.2 Stable Homotopy Theory

We easily see that the (de-)suspension of a suspension spectrum of aCW-complex is a CW-spectrum, further if it has only finitely many cellsthen its suspension spectrum is finite, and if it has only finitely many cellsof each dimension the its suspension spectrum is of finite type. We see thata strict map of CW-spectra is a CW map. There is some work involved inshowing that the definition of a CW-map is indeed well defined, but this willnot be done in this thesis.

CW spectra can, as opposed to CW-complexes, have cells of negativedimension.

Example 1.1.8. We wish to define a spectrum X with Xn := S1 ∨ S2 ∨ . . .for all n, now since the suspension distributes over wedge sums, we takethe structure maps to be the obvious inclusions, then this is clearly a CW-spectrum and further it has a cell in all dimensions.

Proof. So for each Xn the non basepoint cells are e1n, e

2n, . . ., and we get from

the structure maps the following identifications

X1 := e10 e2

0 e30 . . .

X1 := e11 e2

1 e31 . . .

X2 := e12 e2

2 e32 . . .

......

......

. . .

Here the diagonal lines denote identifications. So for en0 then en+kk is a cell

of Xk for all k and hence it is a cell of dimension n. On the other hand e1n

is a cell of Xn and ek−n+1k is a cell of Xk for all k ≥ n+ 1, so e1

n representsa cell of dimension −n+ 1.

Given a CW-spectrum X we can define the n’th-skeleta of X to be the

CW-spectrum X(n) where (X(n))k = X(k+n)k with the obvious structure

maps, where X(k+n)k is the k + n’th skeleta of Xk as a CW-complex. Given

a CW-complex X we see that (Σ∞X)(n) = Σ∞(X(n)).

1.2 Stable Homotopy Theory

It is possible to equip the category of spectra with a model category struc-ture. We will not go into great detail with this, but only note some of thedetails regarding the definition of the weak equivalences. For this we willneed the Q-construction:

4


Lemma 1.2.1. Let X be a spectrum then there is a spectrum QX such that(QX)k is weakly equivalent in the category of pointed topological spaces tocolimjΩjXk+j and the structure maps

ωk : (QX)k → Ω(QX)k+1

are weak equivalences in the category of pointed topological spaces.

Note that this is a functorial construction. Further it always comesequipped with a morphism X → QX.

Definition 1.2.2. Given f : X → Y a strict map (or a CW-map) of (CW)spectra we call it a weak equivalence if it induces levelwise weak equivalencesfk : (QX)k → (QY )k as spaces.

Recall that ifX ′ is a cofinal subspectrum ofX then the inclusionX ′ → Xis a weak equivalence and likewise is X → QX. We will as it is standardwrite X ' Y if there is a weak equivalence f : X → Y . We will for therest of this thesis only work in the homotopy category which arises from thismodel structure.

We note the following easy results:

Proposition 1.2.3.

• For all k, l ∈ Z and spectra X then ΣkΣlX ' Σk+lX.

• For all k ∈ N and pointed spaces X then ΣΣ∞X ' Σ∞(ΣX).

Definition 1.2.4. We will write [X,Y ] for the morphisms between X andY in the homotopy stable category.

This comes equipped with an abelian group structure in a functorialway. This comes from the fact that for any spaces X,Y , [Σ2X,Y ] comesequipped with an abelian structure and we can think of a spectrum as beingthe generalization of a space and all its suspensions. Further recall that forall k ∈ Z then [X,Y ] ∼= [ΣkX,ΣkY ] via α 7→ Σk(α). For this reason we willnot distinguish between a morphism and its (de)suspensions.

Definition 1.2.5. Let X be a spectrum. We define its n’th homotopy group,πn(X), to be [Sn, X].

Note that we allow n to be negative. We remind our selves of the fol-lowing well known results, tying the stable homotopy theory to the notionof stable homotopy groups and Freudenthal Suspension Theorem.

5


Proposition 1.2.6.

• π∗X ∼= colimkπunst∗+kXk where πunst∗ refers to the standard unstable topo-

logical homotopy groups.

• f : X → Y is a weak equivalence if and only if π∗(f) is an isomor-phism.

As with spaces the level of connectivity is of interest to us. Opposed tospaces, being 1-connected holds no great value for our spectra. This notionis replaced with the notion of a spectrum being n-connected for some valueof n.

Definition 1.2.7. A spectra X is called n-connected if πkX = 0 for allk ≤ n. We say that X is connective if X is n-connected for some n.

Now, we want to discuss some constructions that we will be needing. IfX,Y are spectra there is a spectrum called the wedge sum X ∨ Y , with forall spectra Z, [X ∨ Y,Z] is naturally isomorphic to [X,Z]⊕ [Y,Z]. Furtherthere is a spectrum called the smashproduct X ∧ Y , which is unique up toweak equivalences, such that X ∧ Y ' Y ∧ X, called the twist morphism,and S0 ∧ X ' X ' X ∧ S0. Both of these constructions are natural inboth X and Y . We can actually give an easy definition of the wedge sum:(X ∨ Y )n = Xn ∨ Yn with the obvious structure maps. There are of courseexplicit constructions of the smash product but going into a discussion ofthose would take up to much space here, so we will leave it out. The smashproduct of spectra relates to suspension precisely in the manner we wouldexpect:

Proposition 1.2.8. Sn ∧X ' ΣnX.

We will now briefly turn our attention towards the results of the extrastructure that the CW-spectra bring to the table. Note that all of these aregeneralizations of similar results for CW-complexes and the proofs mirrorthe space-level proofs. As stated earlier, spectra in many way conform toour expectations from our intuition of spaces, e.g., we still have the resultthat CW-maps are enough when working with CW-spectra, and that thesemaps respect the cellular structure.

Proposition 1.2.9. Any morphism between X and Y , CW-spectra, is ho-motopic to a CW-map.

Corollary 1.2.10. If X,Y are CW-spectra, where Y has a top cell of di-mension m, it follows that [Y,X] ∼= [Y,X(m+1)] i.e. that up to homotopy allmorphisms hit the m+ 1’th skeleta of X.

6


Proof. Take f : Y → X then we know it is homotopic to a CW map, soassume without loss of generality that it is a CW map and let Y ′ be acofinal CW-subspectrum of Y such that f ′ : Y ′ → X is strict. Then Y ′k hasits highest dimensional cell in at most k +m. So f ′0 is homotopic to a map

f0 : Y ′0 → X(m+1)0 , so by homotopy extension property, and the fact that

f ′ : Y ′ → X was strict we can find f : Y ′ → X(m+1) homotopic to f ′, andnow we are done.

From the above corollary we get the following two useful results:

Corollary 1.2.11. If X is CW-spectrum then πiX ∼= πiX(i+1).

Corollary 1.2.12. If X,Y are CW-spectra, where X has a bottom cell ofdimension n and Y has a top cell of dimension m, then if m < n it followsthat [Y,X] = 0 i.e. that all morphisms are trivial.

Note that this implies that a CW-spectrum with cells only of dimensionn or greater is (n− 1)-connected.

We can also define the mapping cone in almost the usual way, whichthen gives rise to a notion of cofibrations, which are very important as theyallow us to form long exact sequences of homotopy groups:

Definition 1.2.13.

• Given a CW map f : X → Y between CW-spectra. There is a CW-spectrum Cf called the mapping cone of f , with a morphisms

Y → Cf → ΣX

where if X ′ is a cofinal subspectrum of X such that f ′ : X ′ → Y isstrict then we define (Cf )n := Yn ∪f ′ CX ′n, i.e. the pointed topologicalmapping cone of f ′n, with the obvious structure maps. This is uniqueup to weak equivalences.

• A sequence Xf // Y // Z // ΣX is called a cofiber sequence,

and Z is called the cofiber of f if there is f ′ : X ′ → Y ′ a CW-morphismand weak equivalences α, β, γ such that

X ′f ′ //

α

Y ′ //

β

Cf ′

γ

// ΣX

Σα

Xf // Y // Z // ΣX

commutes.

7


We will usually just write a cofiber sequence as X → Y → Z, thusomitting Z → ΣX from our notation. The notion of a cofiber sequence iseven more helpful in the stable setting than in the usual topological settingas it gives rise to long exact sequences continuing in both directions. Wewill now collect some results regarding cofibrations that will be very usefulto us .

Proposition 1.2.14.

• ΣCf ' CΣf

• Let X → Y → Z be a cofiber sequence then for any spectrum E,

X ∧ E // Y ∧ E // Z ∧ E

is a cofiber sequence.

• Let X → Y → Z be a cofiber sequence then any three consecutivemorphisms of the following five:

X // Y // Z // ΣX // ΣY

defines a cofiber sequence.

• If f : X → Y is any morphism then there is some spectrum Z, uniqueup to weak equivalences, and morphisms Y → Z and Z → ΣX such

that Xf→ Y → Z is a cofiber sequence.

• Let Xf→ Y

g→ Z be a cofiber sequence then for any spectrum E thefollowing is exact

[E,X]f∗ // [E, Y ]

g∗ // [E,Z]

[Z,E]g∗ // [Y,E]

f∗ // [X,E]

Note that this means that we get a long exact sequence in stable homo-topy groups from a cofiber sequence.

Corollary 1.2.15. Let X → Y → Z be a cofiber sequence, then there is along exact sequence in homotopy

. . . // πk+1Z // πkX // πkY // πkZ // . . .

Proof. Take E = Sk above and recall that for all spectra A,B we have[A,B] = [ΣkA,ΣkB] for all k ∈ Z

8


1.2.1 Homology and cohomology theories

To further illuminate the similarity with our usual homotopy theory we willdefine some easier to calculate algebraic invariants in the form of (co)homologytheories:

Definition 1.2.16. Given any spectrum E, we can for any spectrum Xdefine the E-(co)homology theory of X by

En(X) := πn(E ∧X)

En(X) := [S−n ∧X,E]

This defines a homology respectively cohomology theory on spectra.

It should be noted that any cohomology theory can be constructed in thisway. Further note that (S0)∗X = π∗X and En(ΣkX) = En−kX. We wouldlike this new notion of (co)homology to on suspension spectra be related tothe usual (co)homology of the space. For this we will need a generalizationof the Eilenberg-MacLane spaces:

Definition 1.2.17. Let K(G,n) be the Eilenberg-MacLane space for thegroup G and n ∈ N then we kan define a spectrum HG, with(HG)n := K(G,n) and the structure maps ωn : HGn → ΩHGn+1 are theusual identifications.

We recall that (HG)n(Σ∞X) ∼= Hn(X;G) and (HG)n(Σ∞X) ∼= Hn(X;G),and therefore for any spectrum X we will write H∗(X) for (HZ)n(X) andlikewise for the cohomology and homology with coefficients.

Note further that proposition 1.2.14 implies that given a cofiber sequencewe get a long exact sequence in E∗ homology for any spectrum E. Likewisewe get a long exact sequence in E∗ going in the opposite direction as wouldbe expected.

We also have a stable version of the Hurewicz map and theorem:

Theorem 1.2.18 (Hurewicz). For any spectrum X there is a natural ho-momorphism

h : πq(X)→ Hq(X)

if X is (n − 1)-connected then this is an isomorphism for q ≤ n and asurjection for q = n+ 1.

We will now recall the definition of a Serre Class in order to generalizethe Hurewicz mod Serre theorem to the stable case:

9


Definition 1.2.19. A class of abelian groups, C, is called a Serre Class ifthe following axioms are satisfied.

1. 0 ∈ C.

2. G ∈ C and H ∼= G implies that H ∈ C.

3. H ≤ G ∈ C implies that H ∈ C.

4. H ≤ G ∈ C implies that G/H ∈ C.

5. If 0→ A→ B → C → 0 is exact and A,C ∈ C then B ∈ C.

Proposition 1.2.20. If X is a (de)suspension of a suspension spectrumwith for all i Hi(X) ∈ C for some Serre Class C then πi(X) ∈ C for all i

Proof. Assume X is ΣkΣ∞Y for some space Y and that for all i Hi(X) ∈ Cfor some Serre Class C. Given arbitrary i we wish to show that πiX ∈ C.Now πiX = colimnπ

unsti+n Σk+nY where the π∗ on the right hand side are the

unstable homotopy groups. Note further that Hi(X,Z) ∼= Hi+k(Y ), whereHi+k(Y ) = 0 when i+ k < 0. Now since Hi+k(Y ) ∼= Hi+k+n(ΣnY ) ∈ C thisimplies that πi+nΣk+nY , again for the unstable homotopy groups, but byFreudenthal this stabilizes at some n and hence πiX ∈ C.

We recall the following Serre Classes of interest:

Example 1.2.21. The following are Serre Classes:

• The class of all finitely generated abelian groups.

• The class of all abelian groups G were for some fixed prime p: Ifx · g = 0 with 0 6= x ∈ Z and 0 6= g ∈ G then x = pi for some i ∈ N.

Lastly we will need the notion of a, not necessarily CW, spectrum offinite type:

Definition 1.2.22. Given a spectrum X we say that it is of finite type ifπ∗X is finitely generated in each dimension.

If X is a spectrum of finite type, then there exists a CW-spectrum XCW

of finite type as a CW-spectrum with a weak equivalence X → XCW. There-fore, we know that if X is of finite type as a CW-spectrum it is of finite typeas a spectrum.

10


1.3 Homotopy limits and colimits

As with spaces we would like to have the notion of limits, however as withspaces the categorical construct of a (co)limit is not strong enough for ourwork in homotopy theory and we therefore require the homotopy (co)limit.We start by defining the homotopy limit.

Definition 1.3.1. Given an inverse system of spectra

X := X0 X1f1oo X2

f2oo . . .oo

we can define the homotopy limit of X, holimiXi, to be the desuspensionof the cofiber of the morphism s :

∏Xi →

∏Xi given by

∏Xi → Xj is

pj − fjpj+1 where the pj is the j’th projection and the minus refers to theadditive structure on [

∏Xi, Xj ].

Now the best we could hope for would be that homotopy limit and ho-motopy groups would commute. Unfortunately this is not the case, butfortunately the failure term is computable and we see that the homotopygroups of the homotopy limit is determined by the limit of the homotopygroups.

Proposition 1.3.2. For any spectrum E, and inverse system of spectra Xas above we have the following short exact sequence

0 // R limiEn+1(Xi) // En(holimiXi) // limiEn(Xi) // 0

here R lim is the first derived functor of the limit.

Now, we will define the homotopy colimit:

Definition 1.3.3. Given a directed system of spectra

X := X0f0 // X1

f1 // X2f2 // . . .oo

we can define the homotopy colimit of X, hocolimiXi, to be the cofiber ofthe morphism σ :

∨∞i=0Xi →

∨∞i=0Xi given by for Xi →

∨∞i=0Xi by id− fi,

where the minus refers to the additive structure of [Xi,∨∞i=0Xi].

Now opposed to homotopy limits, homotopy colimits and homotopygroups do in fact commute, implying that:

Proposition 1.3.4. Let E be any spectrum, and X a directed system ofspectra as above we have the following natural isomorphism

colimiE∗(Xi) ∼= E∗(hocolimiXi)

11

Section 1.4 Localizations

1.4 Localizations

In most of this thesis we will not be working with the spectra as they aredefined above, since the homotopy groups are simply too complicated tocompute. Instead we will be focusing on some of the data found in thehomotopy groups. Localization presents a theoretical approach to removingsome of the data to make some computations possible. This section is takenfrom [Bou79].

The first notion we will need is the notion of a localized spectrum. Thatis a spectrum where we only ”see”what a particular homology theory detectsof the space.

Definition 1.4.1. Given a spectrum E:

• We say that a morphism of spectra, f : X → Y , is an E-equivalenceif f∗ : E∗X → E∗Y is an isomorphism.

• A spectrum A is called E-acyclic if E∗A = 0.

• We say that a spectrum C is E-local if each E-equivalence f : X → Yinduces bijection f∗ : [Y,C]→ [X,C] or equivalently [A,C] = 0 for allE-acyclic spectra A.

• We say that XE is an E-localization of X if there is a morphismf : X → XE which is an E-equivalence, XE is E-local, and givenanother E-local spectrum Z and morphism g : X → Z then there is aunique morphism h : XE → Z such that hf = g.

Note that E-localizations are unique up to weak equivalence by theiruniversal property. Some elementary results are also needed:

Proposition 1.4.2. For all spectra X and E there exists a spectrum XE

with morphism X → XE, such that XE is an E-localization of X. Therefurther exists a E-acyclic spectrum EX such that

EX // X // XE

is a cofiber sequence. Further we know the following for all k ∈ Z and anyspectra X, Y .

• (X ∧ Y )E ' XE ∧ YE.

• (ΣkX)E ' ΣkXE.

• If X → Y → Z is a cofiber sequence then so is XE → YE → ZE.

We note that (−)E defines a functor from spectra to spectra. We are notinterested in all manner of localizations but only some few relating to theprime 2. The following result comes from [Bou79] combining his proposition2.5 with his theorem 3.1 and the fact that HG is a connected spectrum with⊕nπnHG ∼= G

12


Proposition 1.4.3. If G = Z/p for some prime p and X is any spectrumthen there is a splitable short exact sequence

0 // Ext(Z/p∞, π∗X) // π∗XHG// Hom(Z/p∞, π∗−1X) // 0

Further if π∗X is finitely generated then π∗XHG∼= Zp ⊗ π∗X.

We will therefore denote XHZ/p by Xp, and call it the p-completion ofX.

Proposition 1.4.4. If G = Z(J) for J a set of primes then

π∗XHG∼= π∗X ⊗G

If we localize with respect toHZ(p) we will call the result the p-localizationof X, and write it as X(p). Likewise we will call the localization with respectto HQ the rationalization and similarly denote it XQ. We see from abovethat π∗XQ ∼= π∗X ⊗Q.

So 2-completion and localization are two different methods of only de-tecting the 2-torsion of the homotopy groups, immensely simplifying theproblem of finding homotopy groups.

1.5 The Adams Spectral Sequence

One tool available to us for computing the 2-completed homotopy groupsis the Adams Spectral Sequence and we will therefore briefly make someremarks on the classical Adams Spectral Sequence (clASS), since we willbe needing information gleaned from the E2 page of this sequence. Thereare several different versions and formulations. Here I have simply chosenone. More information about the spectral sequence can be found in severalplaces, including [Ada95].

Theorem 1.5.1 (clASS). If X is a spectrum of finite type then there is aspectral sequence with Es,t2 = ExtsA∗(H

∗(X,Z/2), H∗(St,Z/2)) and differen-

tials dr : Es,tr → Es+r,t+r−1r converging to πt−sX2, here A∗ is the Steenrod

Algebra.

When we draw the Er-page of clASS: We select as the horizontal axist − s and as the vertical axis s. We have from Bruner some computationsof the E2 page of stunted real and complex projective spaces, which can befound in the appendices, and these are of great use to us.

The zero’th line we find to be E0,t2 = HomA∗(H

∗(X,Z/2), H∗(St,Z/2))and it detects elements of πtX by the morphism

πtX → HomA∗(H∗(X,Z/2), H∗(St,Z/2))

given by [f ] 7→ H∗(f). For connective spectra there is a finiteness theoremwhich guaranties that above a certain diagonal all elements are related viah0 to elements below the diagonal.

13

Section 1.6 S-Duality

Theorem 1.5.2. If X is a (n − 1)-connected spectrum and A(k) is the

exterior algebra on sq1, . . . , sq2k then

ExtsA∗(H∗(X,Z/2), H∗(St,Z/2)) // ExtsA(0)(H

∗(X,Z/2), H∗(St,Z/2))

is an isomorphism when t− s < n+ s.

In many cases this makes everything above the line t− s = n+ s eitherdisappear or at least become easily computable, ensuring that we can getour results by finite calculations.

In Bruner’s pages some of the module structure is recorded. So a hori-zontal line is multiplication by h0 (which detects 2), diagonal lines is multi-plication by h1 (which detects the generator of π1S

0) and the dashed linesare multiplication by h2 (which detects the generator of π3S

0). AppendixA and B contain sections of the spectral sequence relating to some of thestunted real respecitvely complex projective spectra. Appendix C containssections of the spectral sequence relating to the sphere spectrum and themod-2 Moore spectrum.

1.6 S-Duality

Here we will recall the fact that the stable homotopy category comes equippedwith a notion of duality, namely the S-dual (also called the Spanier-Whitehead0-dual). The following comes from [Swi02]

Definition 1.6.1. Let X,X ′ be finite CW-spectra, then we say that

u : X ∧X ′ → S0

is an S-duality map if uk,n : [Sk, Sn ∧X]→ [Sk ∧X ′, Sn], given by for anyf : Sk → Sn ∧X, then uk,n(f) is the composition

Sk ∧X ′ f∧1→ Sn ∧X ∧X ′ 1∧u→ Sn ∧ S0 ∼= Sn

and uk,n : [Sk, X ′ ∧ Sn]→ [X ∧ Sk, Sn], given by for any f : Sk → X ′ ∧ Sn,uk,n(f) is the composition

X ∧ Sk 1∧f→ X ∧X ′ ∧ Sn u∧1→ S0 ∧ Sn ∼= Sn

are isomorphisms for all k, n ∈ Z. We say that X is an S-dual of X ′.

It is well known that if X is a finite CW-spectrum then a S-dual alwaysexists. Being S-dual is even stronger than it might first appear as there arefurther maps for other spectra than the sphere spectra.

Proposition 1.6.2. If X,X ′ are S-dual then DU,V : [U, V ∧X]→ [U∧X ′, V ]and DU,V [U,X ′ ∧ V ]→ [X ∧ U, V ] are isomorphisms for all spectra U, V .

14


Further we can also talk about the dual of morphisms between spectra:

Lemma 1.6.3. If X,X ′ and Y, Y ′ are S-dual and f : X → Y is any mapthen there is up to homotopy a morphism f∗ : Y ′ → X ′, given by

[X,Y ]∼= // [Y ′ ∧X,S0] [Y ′, X ′]

∼=oo

Some elementary results regarding this duality are

Proposition 1.6.4.

• If X ′ is S-dual to X via µ then X is dual to X ′ via µ τ whereτ : X ′ ∧X → X ∧X ′ is the twist morphism.

• If X ′ and X ′′ are S-dual to X then X ′ and X ′′ are homotopic equiva-lent.

• If X ′ and X, Y ′ and Y and Z ′ and Z are S-dual and f : X → Y andg : Y → Z are any morphisms if f∗ and g∗ denotes the dual maps tof and g respectively then (f∗)∗ ' f , (g f)∗ ' f∗ g∗ and (id)∗ ' id.

• (Σnf)∗ ' Σ−nf∗

• If X,X ′, Y, Y ′ and Z,Z ′ are S-dual and X → Y → Z → ΣX is acofiber sequence then so is Z ′ → Y ′ → X ′ → ΣZ ′, where the mor-phisms are the dual of the morphisms of the original cofiber sequence.

The second to last bullet above ensures that the duality functor takescommutative diagrams to commutative diagrams. As the spheres are ofspecial interest to us we will be needing the duals relating to the spherespectrum. Fortunately these are rather easy.

Proposition 1.6.5. The S-dual of St is S−t and given a morphismα : St → Ss then the dual of α is Σ−t−s(α) : S−s → S−t.

15

Chapter 2

Geometry

In this chapter we will discuss the basic ”geometry” of the stunted projectivespectra, which we will be needing for most of the thesis. The purpose of thisis to get information about the sphere spectrum or at least the 2-completedversion via The Kahn-Priddy Theorem, which gives a link between the ho-motopy groups of RP∞ and S0. The existence of this link exists is not sosurprising, when we recall that one way of constructing RPn is quotientingout the free Z/2 action of Sn. We will start by studying three different con-structions of the topological version of the stunted projective spaces and thensee that these all generalize to the stable category, where we can then gener-alize the notion. We will also briefly touch on the morphisms ι : S−1 → P−kas this is the morphism which Jones’ Kahn-Priddy Theorem is about.

2.1 Spaces

We will start with a brief discussion of the standard topology of stunted pro-jective spaces linking them to Thom spaces and the quadratic construction.We will refer readers elsewhere for a more in-depth treatment.

The most basic and intuitive description of the stunted projective spaceswill be presented first.

Definition 2.1.1. Let Pn+kn = RPn+k/RPn−1 for all n, k ∈ N0 with base-

point the class of RPn−1.

Here we use the convention that RP−1 = ∅ and X/∅ is X with a disjointbasepoint, X+, for all spaces X. We see that we have pointed continuousmaps i : Pn+k

n → Pn+k+1n induced by the inclusion RPn+k → RPn+k+1 and

collapse map c : Pn+kn → Pn+k

n+1 induced by the quotient.

Definition 2.1.2. Pn := colimkPn+kn

Clearly Pn = RP∞/RPn−1. We can trivially get c : Pn → Pn+1 inducedagain by the quotient. Further we have the pinch map π : P k0 → S0 bycollapsing RP k to the non-basepoint.

16

CHAPTER 2 GEOMETRY

2.1.1 Thom Spaces

Another way to define the stunted projective spaces can be given via Thom-spaces. The following definition of and results on Thom-spaces are takenfrom [Hus66], and readers are refered there for proofs and a more in-depthtreatment. Let us start with some definitions of vector bundles. The goal ofthis section is to a vector bundle to associate a space, in form of the Thomspace. This will in some sense generalize the suspension of a space allowingfor ”twisted suspensions”.

Definition 2.1.3. A (real) n-dimensional vector bundle ζ, over X, is apair of non-pointed spaces with a map p : E → X, such that p−1(x) is ann-dimensional real vectorspace for all x ∈ X. Further for all x ∈ X theremust exist x ∈ U ⊂ X open and a homeomorphism ϕU making the followingdiagram commute:

p−1(U)ϕU //

p

U × R

π1yy

U

Further if U, V are two such open subsets belonging to x, y ∈ X respectivelythen ϕU |p−1(U∩V ) = ϕV |p−1(U∩V ). We call X the base space and E the totalspace. We will in most cases not differentiate between ζ and E, if both Xand p are obvious.

When n = 1 we call ζ a linebundle. We will list the examples we willbe needing, most of which pertain to RP k, and we will therefore constructsome notation for this. Recall that one definition of RP k is

L ⊂ Rk+1|L is a one-dimensional subspace

Example 2.1.4. Except where otherwise noted these are vector bundles overRP k.

• We will write nε for the trivial bundle of dimension n i.e.

π1 : X × Rn → X

If X = RP k, we will write nεk.

• We will write γk for the canonical line bundle, so with slight abuse ofnotation we will write γk := (L, v) ∈ RP k × Rk+1|v ∈ L, and thenthe vector bundle becomes π1 : γk → RP k. This is a one dimensionalvector bundle

• We will write γ⊥k for the vector bundle

(L, v) ∈ RP k × Rk+1|v ∈ L⊥

This is a k dimensional vector space.

17

Section 2.1 Spaces

• If X is a smooth manifold, then we have the tangent bundle τ(X).This is a vector bundle. We will write τk for τ(RP k).

As always we will need morphism of our new objects. These shouldrespect the vectorspace structure of the fibers in the obvious way.

Definition 2.1.5. Given two vector bundles p : E → X and q : F → Y . Amap of vector bundles h is two continuous maps H : E → F and h : X → Ymaking the obvious diagram commute, and H restricted to the fiber of a pointis a linear map.

We say that two vector bundles are isomorphic if both these maps arehomeomorphic. This defines an equivalence relation. Given a space X wedefine V ect(X) to be the set of isomorphism classes of vector bundles overX.

Further we would like to be able to add vector bundles to form newbundles:

Definition 2.1.6. Given a vector bundle p : E → X and a map f : Y → Xthen taking the pullback over the obvious diagram gives a vector bundle overY called f∗(E).

Let p : E → X and q : F → X then E × F is a vectorbundle overX ×X. We define the Steifel-Whitney sum E ⊕ F to be the pullback alongthe diagonal map ∆ : X → X ×X.

It is known from K-theory that this operation turns V ect(X) into amonoid and using the Grothendieck construction we get the group K0(X).given a vectorbundle ζ over X, we call −[ζ] ∈ K0(X) a virtual bundle overX. A last definition before we are ready to define what we will mean withthe Thom space of a bundle is the notion of a metric on a vector bundle.This should generalize the inner product of a vector space:

Definition 2.1.7. A riemannian metric on a vectorbundle ζ over X is amap m : ζ ⊕ ζ → R such that m|p−1(b) is a inner product. Given such ametric we can define the disk bundle:

D(ζ) := (x, z) ∈ ζ|m(z, z) ≤ 1

and sphere bundle:

S(ζ) := (x, z) ∈ ζ|m(z, z) = 1

called the disk- respectively sphere bundle.

Clearly all the bundles over RP k considered above allow a riemannianmetric as we can embed RP k → RN for a suitable N , and then, e.g.,γk → RN+k+1 by sending the vector space part to the additional coordi-nates and then we can simply use the usual inner product. We are nowready to define the Thom spaces:

18

CHAPTER 2 GEOMETRY

Definition 2.1.8. Given a vector bundle ζ over a compact space X, wedefine the Thom-space of ζ, Thom(ζ), to be the one point compactificationof the total space, with the added point as basepoint.

Clearly this depends only on the isomorphism class of ζ. Another de-scription can be given for bundles with a metric.

Lemma 2.1.9. If ζ has a riemanian metric then Thom(ζ) = D(ζ)/S(ζ)

Proof. We clearly see that ζ is homeomorphic to D(ζ)−S(ζ), so taking theone-point compactification on both sides proves the lemma

Earlier I called Thom spaces ”twisted suspensions” and the explanationis as follows:

Proposition 2.1.10. If X is compact and ζ is a vector bundle over X thenThom(ζ ⊕ nε) = ΣnThom(ζ).

So if we take ζ to be the trivial bundle X × 0 then we exactly get thesuspension of X back again, up to some basepoint problems. Further thecell structure fits with the cell structure of suspensions to a certain degree.

Proposition 2.1.11. If X is a compact CW-complex with cellsenαα |α ∈ A∪e0, where enαα is an nα-cell and γ is a k-dimensional vector-bundle over X, then Thom(γ) is a CW-complex with cellsenα+kα |α ∈ A ∪ e0.

So we can see that the cells corresponds to those of a suspension ofthe basespace. We are now ready to explain how we construct our stuntedprojective spaces with Thom spaces. We simply take the Thom space of ourfavorite vector bundles over RP k:

Proposition 2.1.12. For k, n ≥ 0: P k+nn = Thom(nγk).

We wish to define maps P k+nn → P k+n+1

n . Take ι : RP k → RP k+1 tobe the canonical inclusion. Now form ι∗(nγk+1). We see that this is nγk.So the induced map also gives us a map on the one point compactification(since it is a linear inclusion on each fiber), and hence we get a map fromThom(nγk) → Thom(nγk+1) which induces a map i : P k+n

n → P k+n+1n ,

which is equal to our usual inclusion.

2.1.2 Quadratic Construction

A different way to construct our spaces is via the quadratic construction.This expresses the stunted projective spaces as the homotopy fix points ofZ/2. This construction will be well known to most readers as it is used inthe construction of the Steenrod squares. This section is taken primarilyfrom [JW83].

19

Section 2.1 Spaces

We recall that Z/2 acts on Sn with the antipodal action and thatSn/Z/2 = RPn. Given a pointed space X we can form a Z/n action onXn by permuation on the coordinates. This gives an action when we passto the smash product instead, so there is an Z/2-action on X ∧X.

Definition 2.1.13. We define the pointed space Dn2X, where X is a pointed

space, to be

Dn2 (X) =

Sn × (X ∧X)

Z/2

/RPn × ∗

where the action on Sn × (X ∧X) is the diagonal one.

This is well defined since Sn injects into Sn× (X ∧X), by s 7→ (s, ∗) and

this map respects the Z/2 action, so we get that RPn injects into Sn×(X∧X)Z/2 .

Clearly the inclusion of Sn into Sn+1 gives a pointed map from Dn2X to

Dn+12 X. We can form the colimit over all of these, and we call this the

quadratic construction on X and write it D2X. We see that given a pointedmap f : X → Y this clearly induces a pointed map D2(f) : D2X → D2Y .

Now let Z be a pointed space as well, then we get a map

ϕZ : Z ∧Dn2X → Dn

2 (Z ∧X)

z ∧ (s, x1 ∧ x2) 7→ (s, (z ∧ x1) ∧ (z ∧ x2))

This is natural in the sense that if we have a map f : Z → Z ′ and g : X → Ywe get the following commutative diagram.

Z ∧D2(X)ϕZ //

f∧D2(g)

D2(Z ∧X)

D2(f∧g)

Z ′ ∧D2(Y )ϕZ′ // D2(Z ′ ∧ Y )

A special case is Z = S1. This gives us a map ΣDn2X → Dn

2 ΣX, which isagain natural. So again we want to get the stunted projective spaces, andwe arrive at these by using this construction on the spheres.

Proposition 2.1.14. Dn2S

k ∼= ΣkP k+nk for all k, n ∈ N0.

Proof. We take ν := Sn×Rk ×Rk → Sn being the 2k trivial vector bundle.Then we can define a Z/2 action on this by the usual action on Sn andpermutation of coordinates on Rk×Rk. We can now switch basis on Rk×Rk,such that the first coordinates are (x, x) and the second are (y,−y). so weget Rk+ × Rk− where the first factor is generated by (1, 1) and the secondby (1,−1) then we see that the action now is trivial on the first factor and

20

CHAPTER 2 GEOMETRY

multiplication by −1 on the second factor. So taking the quotient by thegroup action we get

Sn × Rk+ × Rk− //

kεn ⊕ kγn

Sn // RPn

Now taking the one-point-compactification gets us the Thom space ofkεn ⊕ kγn, which is ΣkP k+n

n , but we also see that taking the one pointcompactification is the same as taking the one point compactification ofSn × Rk × Rk and then quotienting out the action which gives us Dn

2Sk.

Since Thom(ν) = Sn × (Rk × Rk)∞/S2 ×∞ = Sn × Sk ∧ Sk/S2 × ∗, thenquotienting out the action gives us what we want.

Clearly D2Sk = ΣkPk by taking the colimit.

So let us now define the same map as before ϕs1 : ΣD2Sk → D2S

k+1 by

s ∧ (t, x ∧ y) 7→ (t, (s ∧ x) ∧ (s ∧ y)

and let’s see what this map does under the identification above. So letn, k ≥ 0 then we see that ΣD2S

k is the one point compactification of

R× R∞ × Rk × Rk

Z/2

by the same basis change as above we see that ϕS1 can be thought of asinduced by

R×R∞ × Rk+ × Rk−

Z/2→

R∞ × Rk+1+ × Rk∗

Z/2

where Rk∗ ⊂ R2k+2 has basis (1, 0,−1, 0), and from this it is now clear thatϕs1 = Σkc, where c is the usual collapse map.

2.2 Spectra

Now that the notion of a stunted projective space is well defined we willturn our attention to a generalized stable version. The goal will be to studystunted projective spectra, Pnk where we allow for both n and k to be anyinteger. The constructions of these will precisely mirror the construction ofthe spaces above.

21

Section 2.2 Spectra

2.2.1 Thom Spectra

The goal now is to expand our definition of Thom spaces. It is clear thattaking the suspension spectrum of a Thom space yields a spectrum, but wecan expand the notion even further to include the notion of virtual bundles.Let’s recall from topological K-theory that

Proposition 2.2.1. Every δ ∈ K0(X) can be written as [E]− [nε], for somevector bundle E and some n ∈ N0.

We will use this fact to define what we mean by the Thom spectrum ofa virtual bundle

Definition 2.2.2. Given a virtual vector bundle −[ζ] ∈ K0(X), where X iscompact, we define the Thom spectrum of −ζ to be

Thom(−ζ) := Σ−nΣ∞Thom(E)

where −[ζ] = [E]− [nε].

To see that this is well defined, assume that [F ]−[mε] = −[ζ] = [E]−[nε],that means that there is G vector bundle over X such that

F ⊕ nε⊕G ∼= E ⊕mε⊕G

Now we know from K-theory that there exists some G⊥ such that

G⊕G⊥ = kε

for some k, so we get that F ⊕ nε ⊕ kε ∼= E ⊕ mε ⊕ kε. Clearly nowmε⊕ kε = (m+ k)ε, so taking the Thom space of this we get

Σn+kThom(F ) = Σm+kThom(E)

Passing to spectra and desuspending both sides by n + m + k we get thatΣ−mThom(F ) ' Σ−nThom(E), which was what we wanted. We see thatγk ⊕ γ⊥k = (k + 1)εk, so Thom(−nγk) = Σ−nk−nThom(nγ⊥k ).

Take i : RP k → RP k+1 to be the canonical inclusion. Then formi∗(nγ⊥k+1), we see that this is nγ⊥k ⊕ nεk. So the induced map also gives usa map on the one point compactification (since it is linear inclusion on eachfiber), and hence we get a map from ΣnThom(nγ⊥k )→ Thom(nγ⊥k+1) which

induces a map of spectra between Σ−nk−nThom(nγ⊥k )→ Σ−nk−2nThom(nγ⊥k+1),which is the map we wanted when n < 0. For n ≥ 0 the same constructionwill give us the usual inclusion.

22

CHAPTER 2 GEOMETRY

2.2.2 Quadratic Construction

Again do we seek to expand our construction to spectras. In this case we dothe obvious thing, i.e., apply our construction almost unchanged to spectra,which turns out to work well:

Definition 2.2.3. Given a spectrum X we can define Dk2X to be the spec-

trum with (Dk2X)n = Sk×(X∧X)n

Z/2 /RP k×∗ and if σ : (X∧X)n → (X∧X)n+1

is the structure map of X ∧X, then idSk × σ induces the structure map ofDk

2X. Likewise we can define D2X in the same manner as before.

Now let X be a pointed space, then D2

(Σ∞(X)

)' Σ∞

(D2(X)

). Fur-

thermore as with spaces we have a natural map for spectra X,Z,

ϕZ : Z ∧D2X → D2(Z ∧X)

which as before gives us c : ΣD2Sn → D2S

n+1, for all n, when Z = S1 andX = Sn. Given α ∈ πtS0 we can define Q(α) : Pt → S−t by desuspending ttimes the following composition

D2(St)D2(α) // D2(S0)

π // S0

This morphism will come in handy when we prove Jones’ Kahn-Priddy The-orem since we will use it in constructing a useful commutative square.

2.2.3 The Spectra Pn

This section is taken from [BMMS86] section V.2. Finally we are now readyto define our notion of stunted real projective spectra. As with spaces wewill give three different constructions which will be useful as each of themhave their own advantages. We define φ : N → N as follows: Let φ(k) bethe number of positive integers i ≤ k such that i is congruent to 0, 1, 2, or 4modulo 8. Then

Definition 2.2.4. For n ∈ Z and k ∈ N0 and for any r ≡ n modulo2φ(k), r ≥ 0. We define the stunted real projective spectra, Pn+k

n , to beΣn−rΣ∞P r+kr .

The reason for this perhaps odd looking definition is to be found withthe James Periodicity. It is known that for spaces that Pn+k

n is homotopicequivalent to Σn−rP r+kr with n ≥ r ≥ 0 and r ≡ n modulo 2φ(k), so thisensures that the definition coincides with what we would expect for n ≥ 0,and that the homotopy structure is correct when n < 0. We also have thefollowing result:

23

Section 2.2 Spectra

Theorem 2.2.5. For all k ∈ N and n ∈ Z

• Dk2S

n ' ΣnPn+kn

• Thom(nγk) ' Pn+kn

• If n ≡ m (mod 2φ(k)) then Pn+kn ' Σn−mPm+k

m for all n,m ∈ Z.

Hence we get from the Thom spectra approach inclusion morphismsi : Pn+k

n → Pn+k+1n and we call the homotopy colimit over these Pn, and see

that this conforms with our previous notation. We get from the quadraticconstruction approach maps c : Pn → Pn+1. We see that when k > n ≥ 0then

P knc→ P kn+1 → Sn+1

is a cofiber sequence since Sn+1 is homotopic to the mapping cone of c. Soby suspending this in a suitable fashion we see that this is a cofiber sequencefor all k, n ∈ Z and hence we also have cofiber sequences

Pnc→ Pn+1 → Sn+1

2.2.4 The CW-structure

It is well known from basic algebraic topology that RP k+n has a cell in eachdimension as a CW-complex and hence for all n ≥ 0, the space P k+n

n isa CW-complex with a cell in each dimension between n and n + k (bothincluded) and an extra cell in dimension 0. So as a spectrum this is aCW-spectrum with one cell in each dimension between n and n + k (bothincluded). When k =∞ Pn has one cell in each dimension greater or equalto n.

By definition 2.2.4 we see that P k+nn again is a CW-spectrum (as it is

the (de)suspension of a suspension spectrum of a CW-complex) with a cellin each dimension between n and k + n again with both included. Now letk+n > 0 then we can easily see that P k+n+1

n −P k+nn is an open stable k+n+1

cell, so Pn must have a stable cell in each dimension from n onwards. Fromthis we infer that Pn is of finite type and connective. Further, we observethat taking the n-skeleta of Pmk for m ≥ n gets us (Pmk )(n) = Pnk .

Note that this further implies by the Hurewicz mod Serre theorem that

Corollary 2.2.6. πiP−k are finitely generated for all i, k ∈ Z.

Proof. Recall that πiP−k ∼= πiPi+1−k . Now, P i+1

−k is the desuspension of sus-pension spectrum and hence by proposition 1.2.20 we are done.

24

CHAPTER 2 GEOMETRY

2.3 Atiyah Duality

We saw earlier that the dual of St was S−t and the dual of a morphismof sphere spectra was itself. This is a very nice property as it allows us toeasily move between a diagram and its dual. It would be advantageous forus to be able to do the same with the stunted real projective spectra and inthis section it will turn out that up to suspension this is indeed true, andwe will use this fact many times throughout the thesis.

We see from [Ati61] theorem 6.1 that the S-dual of

T((−n)γk−1

)= P k−n−1

−n

is

T((n− k)γk−1 ⊕ ε

)= ΣPn−1

n−k

So specifically the dual of P k−10 is ΣP−1

−k .

Definition 2.3.1. We define the map ι : S−1 → P−k by desuspending thedual of the pinchmap P k−1

0 → S0 followed by the inclusion P−1−k → P−k

Notice that this map is not trivial whenever k > 0 since the pinch mapis non-trivial in homology. We wish to show that

S−1 ι→ P−1c→ P0

is a cofiber sequence. We notice that P 00 = S0 and the pinchmap is the

identity, so the dual S−1 → P−1−1 which must be a homotopic equivalence by

the duality, so the mapping cone of ι : S−1 → P−1 cones off the bottom celland hence we get the cofiber sequence we want. As usual we can extend it,which gives us

S−1 ι→ P−1c→ P0

τ→ S0

The Kahn-Priddy Theorem concerns itself with this morphism τ .Taking i : P k−1

−n → P k−n to be the inclusion, then we get the followingcofiber sequence

Sk−1 φ // P k−1−n

i // P k−n // Sk

Where we can take φ to be the attaching map of the k-cell of P k−n. Dualizingthis gives the following cofiber sequence.

S−k Pn−1−k

oo Pn−1−k−1

i∗oo S−k−1ψoo

where i∗ is the dual of i. Now by taking the long exact sequence in homotopygroups we see that the morphism, ψ, is a generator for π−k−1P

n−1−k−1. Further

25

Section 2.4 clASS

note that since Pn−1−k−1 and S−k−1 are CW-spectra then we can assume that

ψ is a CW morphism and hence we can form the mapping cone Cψ. Thuswe get the following commutative diagram

S−k Pn−1−k

oo Pn−1−k−1i∗

oo S−k−1ψoo

S−k Cψoo

'

OO

Pn−1−k−1

oo S−k−1ψoo

But since ψ was a generator then we see that up to choice of generatorsthis exactly recovers the collapse map. So we have now proved the followinghandy lemma:

Lemma 2.3.2. The S-dual morphism of the inclusion P k−1−n → P k−n is the

collapse map c : Pn−1−k−1 → Pn−1

−k .

2.4 clASS

Our primary method of getting information about the stable homotopygroups of spectra is the classical Adams Spectral Sequence, and thereforewe will also briefly state a result relating our morphism ι : S−1 → P−k tothe spectral sequence so that we may glean information pertaining to thismorphism in the spectral sequence. The E2 pages for some of the P−k’s canbe found in appendix A.

We need to assure our selves that the pages of the appendix do notmislead us. There might be data for very large values of s not in the printedsection that might mess up our calculations. We recall theorem 1.5.2 statingthat above a certain line it is suffices to calculate the Ext-groups over A0.Recall from [BMMS86] that H∗(P−k;Z/2) = Z/2[x, x−1]/x−k−1 with |x| = 1where sqixj =

(ji

)xi+j . This implies that H∗(P−k;Z/2) is free as a A(0)-

module whenever k is odd and hence that Es,t2 = 0 whenever t − s < s − kand s > 0. When k is even then H∗(P−k;Z/2) is the sum of a free objectand Z/2 in dimension −k as an A0 module. This implies that Es,t = Z/2 forall s, t such that t− s = −k and Es,t = 0 when t− s < s−k and t− s 6= −k.Hence, we are now assured that the sections found in our appendix are notmisleading.

The cofiber sequence S−1 ι→ P−1c→ P0 gives in mod 2 cohomology (we

will suppress Z/2 from our notation), H−1S−1 → H−1P−1 → H−1P0 = 0 sothen since H−1S−1 ∼= H−1P−1

∼= Z/2 and hence ι : S−1 → P−1 is non trivial

26

CHAPTER 2 GEOMETRY

in mod 2 cohomology. Then we have the following commuative diagram:

S−1 ι //

ι

""

P−1

P−k

c

OO

for all k > 0. So this implies that ι : S−1 → P−k is non trivial in cohomologyfor all k > 0. Hence, we know that ι is detected up to some extensionproblem in t = −1 and s = 0. But since all of these have finitely many dotsthen some odd multiple of ι must represent the bottom s = 0 line, but then2iι is detected by the i’th dot in t− s = −1 line.

27

Chapter 3

Jones’ Kahn-PriddyTheorem

We have now come to the point were we are ready to prove Jones’ Kahn-Priddy Theorem. The proof closely follows the one given by Miller [Mil90].

The goal of the theorem is to try to compute the stable homotopy groupsof the sphere spectrum by linking it to the stable homotopy groups of theinfinite real projective space in the hope that the latter are easier to calculate.The proof itself is rather short and elegant. After the proof we will turn ourattention to various applications of the theorem, namely in this chapter theMahowald filtration and root invariants. Both of these constructions closelyrelate to The Kahn-Priddy Theorem, as they are different sets which logwhich elements of the stable homotopy groups of the sphere are detected bywhich stunted projective spectra.

3.1 The proof

We will start by the following small lemma, which is crucial to the proof ofthe theorem.

Lemma 3.1.1. Take α ∈ πtS0, c(r) : P−r → P0 to be the composition of thecollapse maps with itself r-times, and π the pinch map, then the followingdiagram commutes for 0 ≤ t < r

P−r

c(r+t)

πc(r) // S0

α

PtQ(α) // S−t

Here Q(α) is the tth-desuspension of D2(St)D2(α)→ D2(S0) = P0

π→ S0.

28

CHAPTER 3 JONES’ KAHN-PRIDDY THEOREM

Proof. Recall the natural morphisms, which came with the quadratic con-struction ϕK : K ∧ D2X → D2(K ∧ X), which were natural with respectto both K and X. We suspend our diagram t times making several of themorphisms clearer to us:

St+r ∧D2(S−r)

α∧1

((

ϕSr //

ϕSt+r

St ∧D2(S0)π //

α∧1

St

α

Sr ∧D2(S−r)ϕSr

((

S0 ∧D2(S0)

∼=

D2(S0)

π

%%D2(St)

Q(α)//

D2(α)

33

S0

where each of the smaller squares can be easily seen to commute, by natu-rality of the ϕ maps, since S0 ' X ' X ' X ∧ S0 in a natural way, andfurther these morphisms are equal to ϕs0 when X = D2Y for some Y . Thisproves the lemma since up to suspension ϕsr is c(r).

We are now ready to state and prove Jones’ strengthening of The Kahn-Priddy Theorem.

Theorem 3.1.2 (Jones’ Kahn-Priddy Theorem). If s < t ∈ Z then for anyα ∈ πt−1S

−1, the composition St−1 α→ S−1 ι→ P−s−1 is trivial.

Proof. Note we can take s ≥ 0 and t ≥ 0 since otherwise the result is trivial.For any β ∈ πt−1S

−s−1 let us study the following diagram

St−1 α //

β

S−1

ι

S−s−1 j // P−s−1

(3.1)

where j is the inclusion of the bottom cell. We wish to determine whenthe diagram commutes, which in general it doesn’t. Clearly the diagramcommutes if and only if it commutes on the (r − 1)-skeleta for r > t. Sodiagram (3.1) commutes if and only if the following diagram commutes:

St−1 α //

β

S−1

ι

S−s−1 j // P r−1−s−1

(3.2)

29

Section 3.1 The proof

A diagram of spectra which each has an S-dual commutes if and only if thedual diagram commutes, this implies that (3.2) commutes if and only if thefollowing diagram commutes:

ΣP s−rπc(r) //

S1

α

Ss+1 β // S−t+ 1

(3.3)

Now (3.3) commutes if and only if the desuspension commutes

P s−rπc(r) //

S0

α

Ss

β // S−t

(3.4)

Now by splicing it with the diagram from our lemma 3.1.1 we get the fol-lowing diagram

P s−rπc(r) //

S0

α

PtQ(α)

!!Ss

β // S−t

The outer square commutes if and only if the lower inner squares does. Since,by lemma 3.1.1 the upper square commutes. Therefore (3.4) commutes if

P s−r

// Pt

Q(α)

Ssβ // S−t

commutes. Now, if s < t and β is trivial then this diagram commutes, sinceP s−r has cells stopping in dimension s and Pt has cells starting in dimensiont so P s−r → Pt is trivial. This implies that when s < t then

St−1 α //

0

S−1

ι

S−s−1 j // P−s−1

commutes and hence St−1 α→ S−1 ι→ P−s−1 is trivial.

30


We will now as corollaries give two more standard formulations of TheKahn-Priddy Theorem.

Corollary 3.1.3 (The Kahn-Priddy Theorem v.1). τ : P0 → S0 is a sur-jection on all positive homotopy groups.

This is perhaps not very surprising as P0 contains a copy of S0, butnevertheless it is convenient to have the corollary.

Proof. We recall the cofiber sequence

S−1 ι→ P−1c→ P0

τ→ S0

which we used to define τ . We recall that πt−2(ι) : πt−2(S−1)→ πt−2P−1 istrivial for 2 ≤ t. Hence from the long exact sequence from a cofiber sequencewe get the following short exact sequences.

0→ πiP−1 → πiP0πiτ→ πiS

0 → 0

for all i ≥ 1. So here πiτ is surjective.

A perhaps more interesting formulation of The Kahn-Priddy theorem isthe following statement. Below τ ′ should be thought of as the restriction ofτ .

Corollary 3.1.4 (The Kahn-Pridy Theorem v2). There exists a morphismτ ′ : P1 → S0 which is surjective on positive 2-localized homotopy groups.

Proof. We start by noting that for any space X: Σ∞X+ is stably homotopyequivalent to Σ∞(X)∨S0. This follows since ΣX+ and Σ(X∨S0) (as spaces)are both non-basepoint preserving homotopy equivalent to the unreducedsuspension of X+, and hence they are homotopy equivalent. This meansthat P0 is homotopy equivalent to P1 ∨ S0 and hence we can write

τ : P1 ∨ S0 ' P0τ→ S0

Now any morphism out of a wedge can always be given as a morphismfrom each wedge summand and further any morphism S0 → S0 is uniquelygiven by its degree so τ = τ ′ ∨ d for some d ∈ Z and τ ′ : P1 → S0. SinceH0(S0) ∼= H0(P1 ∨ S0) and π0P0 → π0S

0 is multiplication by 2 we seethat d = ±2. Recall that πiS

0 → πiS0(2) is surjective when i > 0, so

πiτ(2) = πi(τ′, 2)(2) is surjective. Below we will prove the following small

lemma which will complete our proof.

Lemma 3.1.5. If A doesn’t have non-p torsion and is finite and there existsτ : B → A such that (τ, p) : B ⊕A→ A is surjective then τ is surjective.

And this proves that τ ′ is surjective on positive 2-localized homotopygroups since τ was surjective on positive 2-localized homotopy groups.

31

Section 3.2 Mahowald filtration and root invariants

Now, we will prove the lemma we needed for the above result:

proof of Lemma 3.1.5. Assume that pnA = 0. Then we have the followingcommutative diagram

pn(B ⊕A)(τ,p) //

_

pnA _

... _

... _

p(B ⊕A) _

(τ,p) // pA _

B ⊕A

(τ,p) // A

Then for all i we see that pi(A ⊕ B)/pi+1(A ⊕ B) → piA/pi+1A is inducedby τ , so given a ∈ A we know that there exists a0 ∈ A and b0 ∈ B suchthat a = τ(b) + pa0, but then there exists b1 ∈ pB and a1 ∈ pA suchthat τ(b1) + pa1 = a0, so carrying on we get b0, . . . , bn ∈ B such thatτ(b0) + 2(τ(b1) + 2(. . .+ 2(τ(bn) . . .) = a, and hence we are done.

Note that the same is true for 2-completed homotopy groups since whenG is a finite abelian group then G⊗ Z(2)

∼= G⊗ Z2.

3.2 Mahowald filtration and root invariants

We will now turn our attention to two applications of Jones’ Kahn-PriddyTheorem: The (real) Mahowald filtration and root invariants. The Ma-howald filtration is a well studied filtration of the stable homotopy groupsof the spheres. It filters the stable homotopy groups of the spheres by howclose P−s−1 is to be the sphere spectrum, as the 2-completion of S−1 is thehomotopy limit of the stunted real projective spectra. The root invariantsfurther tie in to the Mahowald filtration as well as the spectral sequencewhich will be set up in chapter 4. Throughout this chapter all homotopygroups and spectra are assumed to be 2-completed. Before proceed to thedefinitions we will be needing the following:

Lemma 3.2.1. The two localization of ι induces a homotopy equivalenceS−1

2∼= holimk(P−k)2

Proof. By [Lin80] we know that ι induces a homotopy equivalence

S−12∼= holimkP−k

32


We define spectra X2k+1 = P−2k−1 and X2k = P−2k−1 with morphismsX2k+1 → X2k (the identity) and X2k → X2k−1 (the collapse map). Furtherwe have morphisms X2k+1

=→ P−2k−1 and the collapse map X2k → P−2k.Since all of these morphisms are compatible we get the following commuta-tive diagram with short exact rows.

0 // R limk π∗+1X−k

// π∗holimkX−k

// limk π∗X−k

// 0

0 // R limk π∗+1P−k // π∗S−12

ι // limk π∗P−k // 0

where ι is induced by ι. Now, since the first and last vertical maps areisomorphisms then so is the middle one. So holimkX−k ' holimkP−k ' S−1

2.

Further we see that the homotopy equivalence S−12→ holimkXk is induced

by the morphism ι. Now take C to be the Serre class of elements withonly 2-torsion. Now for all n it holds that Hn(Xk) ∈ C so πn(Xk) ∈ Cbut that implies that π∗Xk ⊗ Z2

∼= π∗Xk so Xk ' (Xk)2. Then by thesame kind of argument as above now with 2-completed spectra we see thatholimk(P−k)2 ' S

−12

via the 2-completion of the morphisms ι.

We are now ready to define the Mahowald filtration:

Definition 3.2.2. We define the sth Mahowald filtration of the tth stablehomotopy groups of the sphere, M sπt, to be

Ker [ι∗ : πt−1S−1 → πt−1P−s]

where we take everything to be 2-completed.

Clearly M s+1πt ⊂ M sπt, so this is a filtration. Trivially we see thatM0π∗ = π∗ since ι : S−1 → P0 was trivial and hence so is its 2-completion.By lemma 3.2.1 above and an argument analogous to the proof of lemma4.2.1 below we know that

⋂sM

sπ∗ = 0 since ι : S−1 → holimtP−t is the2-adic completion, and R lims π∗(P−s)2 = 0. Further from Jones’ KahnPriddy-Theorem we know that:

Corollary 3.2.3. M sπt = πt for s ≤ t.

Proof. Jones’ Kahn-Priddy theorem states that morphisms induced by ι,πt−1S

−1 → πt−1P−s, here the non completed homotopy groups, are trivialwhen s ≤ t, so taking completion amounts to tensoring the homotopy groupswith Z2, and hence the result follows.

So now that for each element of πt we can define its root invariant by firstnoting where in the Mahowald filtration it appears first and then observehow it relates to the inclusion of the bottom cell.

33


Definition 3.2.4. Given α ∈M sπt −M s+1πt we define the Mahowald rootinvariant (or simply root invariant) of α, R(α), to be the set of morphismsβ : St−1 → S−s−1 making the following diagram commute

Ss−1

k

##St−1

β;;

α

##

P−s−1

S−1

ι;;

We know that R(α) is well defined for all α ∈ π∗ as⋂sM

sπt = 0 byLin’s Theorem and we also know that R(α) is non empty since we know alift β to exist in the situation below by the exactness of cofiber sequenceson homotopy classes.

Ss−1

k

##St−1

=0 BB

β;;

α // S−1

ι

##

ι // P−s−1

c

P−s

We easily see that

Lemma 3.2.5. α ∈ R(α) if and only if α is a generator of π0.

Proof. α ∈ R(α) obviously implies α ∈ M0πt − M1πt. We recall thatM0πt = πt, so we need to study the candidates that are not in M1πt. Werecall that π−1P−1

∼= Z/2 and π−1(ι) is a surjection, so π−1P−1 is generatedby ι. Then clearly if α is multiplication by 2i for some i ≥ 1 then ι α = 0.Now since ι : S−1 → P−1 is the inclusion of the bottom cell then any nontrivial composition lifts as it self, but those are exactly all the uneven el-ements of π−1S

−1, which are also all generators for the 2-completion andhence we are done.

We now turn our attention to calculation of some of the simpler rootinvariants. In this effort we are heavily supported by Bruner’s calculationsfound in appendix A.

Lemma 3.2.6. Let η ∈ π1, ν ∈ π3 and σ ∈ π7 be the standard Hopffibrations and for j ∈ Z let j ∈ π0 be multiplication by j : S0 → S0. Thenη ∈ R(2), η2 ∈ R(4), η3 ∈ R(8), ν ∈ R(η), ν2 ∈ R(η2) and σ ∈ R(ν).

34


Proof. We recall that (2k + 1)ι is represented by the dot in E0,−12 in clASS

for P−k and that the dot in E0,−k2 represents the map S−k → P−k. We define

x : S−1 → P−k to be the generator for the cyclic summand of the homotopygroup containing ι, then we know that (2n+1)x = ι for some n ∈ Z. We willnow investigate the first possibility of the composite ια : St → P−k beingnon trivial, where α = 2, 4 8, η, η2 or ν. So by reading off from Bruner’swe get that

• 2x : S−1 → P−2 is non trivial since there exists a non trivial elementin E−1,1

2 and the only possibility for this to go away is if it has anexiting differential, but since π−2P−2 = Z this cannot happen. Nowby the above we see that 2ι = 4nx+ 2x = 2x since π−1P−2

∼= Z/4.

• By the long exact sequence in homotopy from the cofiber sequenceP−3 → P−2 → S−2 and the fact that π1P−3 = π0P−2 = 0 we get ashort exact sequence

0→ π1P−2 → π0S−3 → π0P−3 → 0

and from the homotopy groups we know we can see that π0P−3 is nontrivial so ιη is non trivial since the generator, x multiplied with η isnon-trivial and ι = (2k + 1)x, so ηι = η(2k + 1)x = ηx since 2η = 0and then it also follows that ι4 is non trivial, since 4x 6= 0.

• Since ιη : S0 → P−3 is non trivial then so is ιη : S0 → P−4 and hence8ι : S−1 → P−4 is non trivial since there is a dot in E−1,2

2 , whichcannot have any entering or exiting differentials.

• We have the cofiber sequence P−5 → P−4 → S−4. Then sinceπ2P−4 = π1P−4 = 0 and π1S

−5 ∼= Z/2 then the long exact sequencein homotopy gives π1P−5

∼= Z/2 but this implies that ιη2 6= 0, sincefor x generator for π−1P−5, η2x 6= 0 and then by an argument simi-lar to the one above η2ι is non trivial and hence so is νx. Now sinceπ2P−5 = Z/2 then 2νx = 0, so since ι = (2k + 1)x we know that νι isnon trivial.

Now since the inclusion of the bottom cell S−k → P−k is a generator forthe −k’th homotopy group we know that it is an uneven multiple of anyother generator, and from this we clearly see that the following diagramscommute:

S−2 // P−2

S−3 // P−3

S−1

0

55

η

==

2 // S−1 ι //

ι==

P−1 S−1

0

55

η2==

4 // S−1 ι //

ι==

P−2

35


S−4 // P−4

S−3 // P−3

S−1

0

55

η3==

8 // S−1 ι //

ι==

P−3 S0

0

55

ν

>>

η // S−1 ι //

ι==

P−2

S−5 // P−5

S−5 // P−5

S1

0

55

ν2>>

η2 // S−1 ι //

ι==

P−4 S2

0

55

σ

>>

ν // S−1 ι //

ι==

P−4

And this proves the lemma.

So there are at least two reasons for the root invariants to be of interestto us. One is, as [Rav84] suggests that, this is an interesting way to constructnew high dimensional elements of the stable homotopy groups of the sphereup to indeterminacy. The other is again related to the spectral sequencethat we will set up next and which will be handled there. It will turn outthat the root invariants of α are the elements in the spectral sequence whichdetects α.

36

Chapter 4

Spectral Sequence

In this chapter we will describe the spectral sequence which arises fromour tower of stunted projective spaces. The spectral sequence is called Ma-howald’s stable EHP sequence and can be found in the literature under thisname. We will follow [Boa99] quite closely to give a detailed construction ofthe sequence. It should be noted that doing computations with the spectralsequence is quite complicated as both the input and the output is the stablehomotopy groups of the sphere, but on the other hand it will turn out thatthe filtration of the target is just a re-indexing of the Mahowald filtration ofwhich we have some knowledge that we can use to our advantage.

4.1 Set-up

We will now start our construction of the spectral sequence. From what wehave already shown we have the following tower

...

c

P−2

c

S−1

ι

@@

ι

77

ι //

ι=0

''ι=0

P−1

c

P0

c...

where c : Ps → Ps+1 is the collapse map and ι : S−1 → P−1−s → P−s has been

defined earlier. Clearly due to connectedness all morphisms ι : S−1 → Ps are

37

Section 4.1 Set-up

trivial for s > −1. From [JW83] 2 · 3 we have the following cofiber sequence

P−sc→ P−s+1

j→ S−s+1

which we can extend as usually to give us:

P−sc→ P−s+1

j→ S−s+1 k→ ΣP−s (4.1)

From the long exact sequence in homotopy groups we see that k is inclusionof the bottom cell. Following [Boa99] we can now define

As,t := πt−s+1P−s+1

Es,t := πt−s+1S−s+1 = πtS

0

and with abuse of notation we will denote

πt−s+1(j) : πt−s+1P−s+1 → πt−s+1S−s+1

πt−s+1(k) : πt−s+1S−s+1 → πt−s+1ΣP−s = πt−sP−s

as

j : As,t → Es,t

k : Es,t → As+1,t

respectively and i := πt−s+1(c). By the long exact sequence in homotopygroups of cofiber sequences this gives us an unrolled exact couple of the form

. . .i // As,t

j

i // As−1,t−1

j

i // . . .

. . . Es,tk

aa

Es−t,t−1k

dd

. . .

We now have a set-up similar to [Boa99] and will follow his conventions. Wedefine

Zs,tr := k−1(Im[i(r−1) : As+r,t+r−1 → As+1,t])

Bs,tr := j(Ker [i(r−1) : As,t → As−r+1,t−r+1])

Es,tr := Zs,tr /Bs,tr

Here i(r) is i composed with i r times, with the convention that i(0) is theidentity. Further we see that the Er’s are well defined, since Imj = Ker k,and clearly Es1 = Es. Given two elements a, b in As+t,t+r−1, such thati(r−1)(a) = i(r−1)(b), we see that their difference is in Ker i(r−1) and hencej (i(r−1))−1 is a well defined homomorphism

Im[i(r−1) : As+r,t+r−1 → As+1,t]→ Es+r,t+r−1r

So now we can define the differential of the spectral sequence

dr : Es,tr → Es+r,t+r−1

to be induced by j (i(r−1))−1 k.

38

CHAPTER 4 SPECTRAL SEQUENCE

4.2 Convergence

The question now becomes whether the spectral sequence converges and inthat case to what. The answers are respectively yes and the 2-completionof S−1.

By Lin’s Theorem [Lin80], ι : S−1 → holimP−t is the 2 completion, whereι is the morphism induced by all the ι’s, and hence we have the followingshort exact sequence

0 // R limAs // π∗(S−12

)ε // limAs // 0

Where ε is the canonical map from π∗holimX → limπ∗X. In the languageof [Boa99] we get

0 // RA∞ // π∗(S−12

)ε // A∞ // 0

Lemma 4.2.1. RA∞ = 0

Proof. Let t be given. Then we wish to study the limit of the sequencestarting with As,t. For r > t every step after As+r,t+r every group is finitesince, when Ps has one cell in each dimension it follows the homology hasfinitely many generators in each dimension and π∗Ps ⊗ Q ∼= H∗(Ps;Q) istrivial unless ∗ = s and s is even. Then we know by an exercise in [Wei95]that the derived limit is trivial and hence we are done.

We want to show that colimAs = 0, i.e. A−∞ is trivial. This is clearsince if we take ξ ∈ As,t = πt−s+1P−s+1 and r = t+ 1, then

i(r)(ξ) ∈ πt−s+1P−s+t+2

but P−s+t+2 is t − s + 1 connected so it is trivial, and hence represents atrivial class in the colimit.

From [Boa99] lemma 5.4 we get the following filtration of A∞ ∼= π∗(S−12

)by

F s,t = Ker [ε : πt−sS−12→ πt−sP−s+1]

the filtration is by the lemma complete Hausdorff and exhaustive. Now notethat since P−2s−1 is 2-completed we can define a morphismι : S−1

2→ holimsP−s as follows: For s odd ι2 : S−1

2→ (P−s)2 ' P−s;

for s even we will use the composition S−12

ι2→ P−s−1c→ P−s. Since these

are comparable morphisms they induce a morphism to the homotopy limit,as wanted. Further we see that the following diagram commutes for all s.

S−1

ι

!!

// S−12

P−s

39

Section 4.2 Convergence

Now since the ι : S−1 → holimsP−s is the 2-completion then take

ι : S−12→ holimsP−s

to be the unique morphism, making the obvious diagram commute so ε aboveis induced by ι and hence with abuse of notation we will denote this as ι aswell.

This is what [Boa99] calls a half-plane spectral sequence with enteringdifferentials since Es,t are trivial whenever t < 0 due to connectedness, andhence only finally many non-trivially differentials will enter Es,tr for all s, tand r. Let us try to compute d1 in the hope that this will ease our way.

Lemma 4.2.2. d1 : Es,t → Es+1,t is trivial if s odd and multiplication by 2if s is even.

Proof. Let s < 0. We see that d1 : Es,t1 → Es+1,t and is induced by

S−s+1 k→ ΣP−sj→ ΣS−s

so it is induces the morphism S−sk→ P−s

j→ S−s, here k is the inclusionof the bottom cell and j is coning of everything but the bottom cell. Thismorphism is determined by its degree, since it is a morphism between spheresof the same dimension. From the exact sequence in homology we get form > k > 1

. . .→ Hk(Pk−11 )→ Hk(P

m1 )→ Hk(P

mk )→ Hk−1(P k−1

1 )→ Hk−1(Pm1 )→ Hk−1(Pmk )→ . . .

Now the homology of P k1 = RP k is well known to be

Hn(P k1 ) =

Z n = 0 or n = k odd

Z/2 0 < n < k, n odd0 else

So we see that when k is odd Hk(Pk) = Z/2 and hence d1 : Es,t1 → Es+1,t1

is trivial whenever s is odd and when k even Hk(Pk) = Z, we see thatH−s(S

−s) → H−s(P−s) is the identity. From the cofiber sequence (4.1) weinfer the following is exact

πkPk //

∼=h

πkSk //

∼=h

πkΣPk−1

∼=h

// πkΣPk

Hk(Pk) // Hk(Sk) Hk−1(Pk−1)

where the vertical maps are the Hurewitcz homomorphism, so from what wesaw above for k even we see that

Z //

∼=

Z //

∼=

Z/2 // 0

Hk(Pk) // Hk(Sk)

40


And hence Hk(Sk)→ Hk(Pk)→ Hk(S

k) is of degree 2, when k is even, andhence d1 : Es,t1 → Es+1,t

1 is multiplication by 2, whenever s is even.

To extend this to s ≥ 0 we notice that by definition 2.2.4 there is somer >> n such that Pn+k

n = Σn−rP k+rr , so this allows us easily to extend the

above to the non-positive degrees as well.

We now wish to show that RE∞ is trivial since this is needed to showconvergence.

Corollary 4.2.3. RE∞ = 0

Proof. We know from [Boa99] that it suffices to show that Es,tr is finite forsome r. This is clear whenever t 6= 0, but when t = 0 then Es,01

∼= Z and d1

is alternating ·2 or trivial, so Es,02 are all either trivial when s even, or Z/2when s odd. In either case it is finite, and hence we are done.

We now wish to show that Es,tr converges to F s,t/F s+1,t+1. Givenζ ∈ Ker [ι : πt−sS

−12→ πt−sP−s+1], we know that the diagram following

commutes:

πt−sS−12

ε

&&

ε // πt−sP−s−r+1

i

i(r−1)// πt−sP−s

πt−sPs−r+2

i(r−2)88

This implies that we get obtain homomorphism

ε : πt−sS−12→ lim

rIm [i(r−1) : πt−sP−s−r+1 → πt−sP−s]

Now assume that α, β ∈ πt−s+1S−s+1 are such that k(α) = ε(ζ) = k(β),

then α− β ∈ Im j. But since

limr

Ker [i(r−1) : πt−sP−s → πt−sP−s+r] = πt−sP−s

it follows that α− β ∈ Bs,t∞ , so k−1 ε is a well defined homomorphism from

F s,t → Es,t∞ . If we assume that ζ ∈ F s+1,t+1 then this trivially maps to 0,so we get a well defined map F s,t/F s+1,t+1 → Es,t∞ .

Theorem 4.2.4. The map described above F s,t/F s+1,t+1 → Es,t∞ is an iso-morphism.

Proof. LetQs := limr Im [i(r−1)As+(r−1) → As]. Then we know from [Boa99]Lemma 5.6 that we have the following short exact sequence

0→ Es∞k→ Qs+1 i→ Qs → 0

41

Section 4.3 The E1 page

since A−∞ = 0 and RE∞ = 0. We saw above that we had a map

F s/F s+1 → Qs+1

and combining [Boa99] lemma 5.6 with [Boa99] lemma 5.9 we get a shortexact sequence

0→ F s/F s+1 → Qs+1 i→ Qs → 0

implying the desired result, since the obvious diagram commutes.

An easy corollary based on the dimension of the target gives us

Corollary 4.2.5. Es,t∞ = 0 when t < s− 1

Proof. This is clear since F s,t = 0 whenever t− s < −1.

To get back the picture of [Mil90], which to some extend spawned thisthesis, take t = q and −s = u.

4.3 The E1 page

Here is an illustration of a section of the E1 page. The non trivial d1-differentials terminating in the section, which we recall are all ·2, are drawnin as the arrows, the horizontal axis is −s and the vertical is t.

t

−s−5 −4 −3 −2 −1 0 1

0

2

4

6

ZZZZZZZ

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

Z24

Z24

Z24

Z24

Z24

Z24

Z24

0000000

0000000

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

42


4.4 Jones’ Kahn-Priddy Theorem and the E∞-page

We can now use Jones’ Kahn-Priddy Theorem to obtain an easy result re-garding our spectral sequence since we can recall that the filtration F s,t issimply a re-indexing of the Mahowald filtration.

Corollary 4.4.1. F s,t = πt−sS−12

when 2(s− 1) ≤ t

Proof. Note that the statement is trivial when t− s < −1

When t − s > −1 then πt−sS−1 → πt−sS

−12

is a surjection, so for any

α : St−s → S−12

we can find β ∈ πt−sS−1 making the following diagram

commute:

St−s

β

α

##S−1

ι""

// S−12

ι||

P−s+1

where the horizontal morphism is the 2-completion. Now, it follows fromJones’ Kahn-Priddy theorem that if 2(s − 1) ≤ t then ι β is trivial andhence so is ι α.

When t − s = −1 and 2(s − 1) ≤ t then 2(s − 1) ≤ s − 1 implying thats ≤ 1, so we wish to show that St−s

α→ S−12

ι→ P−s+1 is trivial, when s ≤ 1,but this is clear since π−1Pk = 0 when k ≥ 0.

Corollary 4.4.2. Es,t∞ = 0 when 2(s− 1) < t

Proof. If 2s − 2 < t then 2s − 1 ≤ t so 2s ≤ t + 1 and therefore2((s + 1) − 1) ≤ t + 1 implying that F s+1,t+1 = πt−sS

−12

, and hence since

Es,t∞ ∼= F s,t/F s+1,t+1 and F s,t ≤ πt−sS−12

we are done.

We can now arrive at the following image of a section of the E∞-page.An X refers to a term which must be trivial for degree reasons, and X refersto a term which must be trivial due to Jones’ Kahn-Priddy Theorem. Corol-lary 3.1.3 implies that F 2,t−1 = πtS

02

for all t ≥ 1, so the corollary implies

that E1,t∞ = 0 for all t ≥ 1. Therefore, we let a X refer to terms which are

trivial due to the classic Kahn-Priddy Theorem. Hence the terms containedwithin in the thick boarder are the only terms which possibly could be non-trivial. As above the horizontal axis is −s and the vertical is t.

43

Section 4.5 Localizations of the spectral sequence

t

−s−5 −4 −3 −2 −1 0 1

0

2

4

6

XXXXXX

XXXXXX

XXXXX

XXXXX

XXXX

XXXXX

XXXXX

4.5 Localizations of the spectral sequence

Here we will try to investigate whether we can gain additional informationfrom localizations of the set-up. Unfortunately homotopy limits do not com-mute with localizations and therefore to some extent we have to start all overagain. Still we will show that 2-completion works out nicely in this case,which is no great surprise as the target is already 2-completed so somehowthe non 2-torsion part of the stunted projective spaces cannot hold greatimportance for the spectral sequence. Further evidence comes from the factthat the E2 page contains only 2-torsion. Recall that the class of 2-torsiongroups is a Serre class C. This means by Hurewicz modulo Serre that if fora spectrum X if Hn(X) ∈ C for all n then πn(X) ∈ C for all n.

Another reason for our focus on 2-completion is to be found in the clASSwhich helps us compute the homotopy groups needed for finding some of thehigher differentials.

Proposition 4.5.1. There exists a spectral sequence with Es,t1 = πt−s+1St−s+12

with differentials dr : Es,tr → Es+r,t+r−1 converging to π∗S02.

Proof. We note the existence of the following cofiber sequence

S−k2→ (P−k)2 → (P−k+1)2

by proposition 1.4.2. We can then define As,t and Es,t as the 2-completionof the groups above. By lemma 3.2.1 we know that A∞ ' π∗S

−12

. A−∞ is

44


still clearly trivial. By our calculation above we see as above that d1 is eithermultiplication by 2 or trivial so Es,t2 is finite for all s, t so RE∞ is still null.We also note that RA∞ = 0 by an analogous argument to the one above.And we can now give the last argument completely following the proof ofTheorem 4.2.4 to show convergence.

It becomes clear that the filtration arising from this spectral sequenceF s,t is exactly the Mahowald filtration up to reindexing i.e. we have

F s,t = M−s+1πt−s+1

4.6 The Root Invariant

Given our 2-completed spectral sequence we could now start to wonder if weare able to say anything as to which elements detect which on the E∞-page.

From above we know thatEs,t∞ ∼= F s,t/F s+1,t+1 = M−s+1πt−s+1/M−sπt−s+1,

so the non trivial elements of Es,t∞ are the elements represented byM−s+1πt−s+1 −M−sπt−s+1.

Proposition 4.6.1. If α ∈M−s+1πt−s+1 −M−sπt−s+1 and β ∈ R(α) thenβ represents a class in Es,t∞ . In fact we will show that β represents the classwhich α represents in F s,t/F s+1,t+1.

Proof. Let the setup be as above. Then we know that the following diagramcommutes:

S−s

k

##St−s

0 ))

β<<

α // S−1

ι

##

ι // P−s

c

P−s+1

Since we assume that ι α : St−s → P−s is non-trivial, then for any r itfollows from the commutivity of diagram below that k β = c(r) ι α andis non trivial

P−s−r

c(r)

St−s

α // S−1ι//

ι;;

P−s

Therefore β ∈ Zs,tr for all r. We have in fact showed that ε(α) = k(β) whichimplies the statement we set out to prove.

45

Section 4.7 Some of the d2’s in the spectral sequence

4.7 Some of the d2’s in the spectral sequence

As all d1’s were alternating multiplication by 2 or trivial, we know that theterms of the E2 page are either the kernel or the cokernel of the multiplicationby 2 map. So from page 2 onwards there is no non-2-torsion. We know thatthe d2 depends on the following diagram

Ps−1//

S−s−1

S−s // P−s

but since our only concern is the 2-torsion part of the spheres we can alsoget away with caring only about the 2-torsion part of the stunted projectivespace. Thus in the following section we can assume that everything is 2-completed. Then we can further use the classical Adams spectral sequenceto calculate some of the homotopy groups of the stunted projective spectra.This further implies that from page E2 onwards the 2-completed spectralsequence is the same as the non-completed spectral sequence. The followingarguments all derive from the same three types of observations: We willuse the diagram above, the fact that the stunted real projective spectra areperiodic, and Bruner’s calculations of the Adams Spectral Sequence as foundin appendix A.

Lemma 4.7.1. d2 : E4n−1,02 → E4n+1,1

2 is an isomorphism for all n ∈ Z.

Proof. We recall that d2 := j i−1 k, so it is induced by taking the π−4n+1

of the following diagram.

P−4nj //

c

S−4n

S−4n+1 k // P−4n+1

Note that from our cofiber sequence we get the following long exact sequencein homotopy

. . . // π−4n+2S−4n+2 k // π−4n+1P−4n+1

i // π−4n+1P−4n+2// . . .

But π−4n+1P−4n+2 is clearly trivial and π−4n+1P−4n+1∼= Z/2 as we saw

when we computed the d1. And by taking a a different long exact sequencefrom our cofiber sequence we get

. . . // π−4n+1P−4ni // π−4n+1P−4n+1

j // π−4n+1S−4n+1 // . . .

46


and hence (since π−4n+1P−4n+1 is finite and π−4n+1S−4n+1 ∼= Z) we infer

that i is surjective.

Now, to complete the argument we need to show that

j : π−4n+1P−4n → π−4n+1S−4n

is surjective. We take the following long exact sequence from our cofibersequence

. . . // π−4n+1P−4nj // π−4n+1S

−4n k // π−4nP−4n−1// . . .

We wish to show that π−4nP−4n−1 is trivial, but we know thatπ−4nP−4n−1

∼= π−4nP−4n+1−4n−1 , and since φ(k) = 2 then also we know from

our geometry chapter that π−4nP−4n+1−4n−1

∼= π0P1−1∼= π0P−1, which we know

from Bruner’s computation with the Adams Spectral Sequence to be trivial.

Now let M(2) be the cofiber of S0 ·2→ S0 (the so called mod-2 Moorespectra), then clearly by periodicity Σ2n+1M(2) ' P 2n

2n+1. Then we canconstruct the following commutative diagram where both rows and columnsare cofiber sequences

P−4n−4n−1

// _

S−4n ·2 //

S−4n

P−4n−1//

c(2)

P−4n

// S−4n

P−4n+1 P−4n+1

// ∗

(4.2)

where ∗ is the trivial spectrum. From appendix C we see that

π−4n+1P−4n−4n−1

∼= Z/4

so by applying π−4n+1 to the diagram above, we get the following commu-

47


tative diagram with exact rows and columns:

...

...

...

. . . // Z/4

// Z/2 0 //

Z/2 // . . .

. . . // Z/2 //

Z/2⊕ Z/2

// Z/2 //

. . .

. . . // Z/2

Z/2 //

0 //

. . .

......

...

Where we read of the groups from appendix A and the periodicity, so sinceπ−2P−3

∼= Z/2 a standard diagram chase reveals that π−4n+1(c(2)) = 0 andhence id ∈ π−4n+1S

−4n+1 lifts to something trivial in P−4n−1. Now, easilywe see that [id] ∈ π−4n+2S

−4n+2, which generates E4n−1,02 is mapped to the

generator of E4n+1,12 .

Lemma 4.7.2. d2 : E4n−1,12 → E4n+1,2


Proof. Since π−4n+2P−4n+2∼= Z, we get from the cofiber sequence

S−4n+1 → P−4n+1 → P−4n+2

that

π−4n+2S−4n+1 → π−4n+2P−4n+1

∼= π−2P−3∼= Z/2

is an isomorphism, where we get the last isomorphism from the periodicityand Bruner’s calculations. From the cofiber sequence

S−4n → P−4nc→ P−4n+1

we get that π−4n+2(c) is surjective since π−4n+2S−4n ∼= Z/2 ∼= π−4n+2P−4n+1

and π−4n+2P−4n∼= Z/2⊕Z/2. So we can now conclude that η lifts to a non

trivial morphism S−4n+2 → P−4n in the following diagram:

P−4n

c

S−4n+2 η // S−4n+1 // P−4n+1

48


We now wish to show that π−4n+2(c(2)) = 0 in the diagram (4.2). So westudy the sequence

. . .π−2(c(2))→ π−2P−3 → π−3P

−4−5 → π−3P−5 → . . .

Now from Bruner’s calculations in appendix A and his calculations in ap-pendix C we find that π−2P−3

∼= Z/2 ∼= π−3P−5 and π−3P−4−5∼= Z/4 implying

that π−2(c(2)) and hence by periodicity so is π−4n+2(c(2)). So by applyingπ−4n+2 to the diagram (4.2) we get from an elementary diagram chase thewanted result.

Lemma 4.7.3. d2 : E4n+1,02 → E4n+3,1

2 is trivial for all n ∈ Z.

Proof. As above we see that this differential is induced by the followingdiagram

P−4n−2j //

c

S−4n−2

S−4n−1 k // P−4n−1

We wish to show that j : π−4n−1P−4n−2 → π−4n−1S−4n−2 is trivial. So we

take the long exact sequence induced by our cofiber sequence and get

. . . // π−4n−2P−4n−3i // π−4n−2P−4n−2

j // π−4n−2S−4n−2 k // π−4n−3P−4n−3

// . . .

Now filling in the groups we know we get that the following exact sequence:

. . . // π−4n−2P−4n−3i // Z // Z // Z/2 // 0

Therefore i must be trivial, so we infer that the following is exact:

. . . // π−4n−1P−4n−2j // π−4n−1S

−4n−2 k // π−4n−2P−4n−3// 0

Now as above we see that

π−4n−2P−4n−3∼= π−4n−2P

−4n−1−4n−3

∼= π−2P−1−3∼= π−2P−3

which we know from Bruner is non trivial, and hence k must be non trivialand since π−4n−1S

−4n−2 ∼= Z/2 it follows the kernel of k is trivial so j mustbe trivial, and hence we are done.

49


Lemma 4.7.4. d2 : E4n−2,12 → E4n,2

2 is trivial for all n ∈ Z.

Proof. Again we study the following diagram

P−4n+1//

S−4n+1

S−4n+3 α // S−4n+2 // P−4n+2// S−4n+2

and ask if α lifts to something non-trivial, but we study the following longexact sequence:

. . . π−4n+3P−4n+1j // π−4n+3S

−4n+1 // . . .

Now, assume for contradiction that j is not trivial, then we get the fol-lowing exact sequence, where the right most zero comes from the fact thatπ−4n+1P−4n+1 is finite and that π−4n+1S

−4n+1 is isomorphic to the integers.

0 // π−4n+2P−4n// π−4n+2P−4n+1

// π−4n+2S−4n+1 // π−4n+1P−4n

rrπ−4n+1P−4n+1

// 0

All of these homotopy groups depend only on their −4n + 3 skeleta andhence this is the same as

0 // π+2P0// π2P1

// π2S1 // π1P0

uuπ1P1

// 0

since φ(3) = 2. But we saw earlier that π1P1∼= Z/2 and we know that

π2S1 ∼= Z/2, so π1P0 can be Z/2, Z/2 ⊕ Z/2 or Z/4. In the first case then

π2P0∼= 0 and in the other two cases π2P0

∼= π2P1∼= Z/2, but from Bruners

data then we know that π2P0∼= π−2P−4

∼= Z/2 ⊕ Z/2 and hence we havearrived at a contradiction.

Lemma 4.7.5. d2 : E4n,12 → E4n+2,2


Proof. Again we study the following diagram

P−4n−1//

S−4n−1

S−4n+1 η // S−4n // P−4n

50


and ask whether a lift exists. Now since we saw earlier that the composition

S−4n → P−4n → S−4n

is multiplication by 2 and 2η = 0 then we know that there is a non triviallift to S−4n+1 → P−4n−1. Now π−4n+1P−4n−1

∼= π1P−1∼= Z/2. And since

π−4nP−4n∼= Z and π−4n+1S

−4n+1 ∼= Z/2 the long exact sequence from thecofiber sequence

. . . // π−4n+1P−4n−1// π−4n+1S

−4n+1 // π−4nP−4n// . . .

implies that π−4n+1P−4n−1∼= π−4n+1S

−4n+1 and hence we are done.

Lemma 4.7.6. For all n ∈ Z. d2 : E8n,22 → E8n+2,3

2 are isomorphisms.

Proof. We start by α : S−8n+2 → S−8n and we wonder when there exists alift S−8n+2 → P−8n−1. Note that S−8n → P−8n → S−8n is multiplicationby two, and since π−8n+2S

−8n = Z/2, we know that there exists some lift toS−8n+2 → P−8n−1. Further since π−8n+2P−8n

∼= π2P0∼= Z/2⊕ Z/2 we infer

that the from the long exact sequence

. . . // π3S1 // π2P0

// π2P1// . . .

that π3S1 → π2P0 cannot be trivial since

π−8n+2P−8n+1∼= π2P1

∼= Z/2

so any α lifts to something non trivial in π−8n+2P−8n−1. We see thatπ−8n+2P−8n−1 → π−8n+2S

−8n−1 is injective since

π−8n+2P−8n−2∼= π−8n+2P

−8n+3−8n−2

∼= π2P3−2∼= 0

by Bruner’s calculations and φ(5) = 3So we now know that α lifts to something non-trivial S−8n+2 → S−8n−1,

so we just need to check that this is not an element in B8n+2,22 . But this is

clearly not the case as j was injective.

Lemma 4.7.7. For all n ∈ Z. d2 : E8n+4,22 → E8n+6,3

2 are isomorphisms.

Proof. Recall that S−8n−4 → P−8n−4 → S−8n−4 is multiplication by 2, soby applying π−8n−2 to the diagram

P−8n−5

c

S−8n−2 η2 // S−8n−4

·2

%%

// P−8n−4

S−8n−4

(4.3)

51


We get by the periodicity and Bruner’s calculations in appendix A the fol-lowing diagram of groups

Z/2⊕ Z/2

Z // // Z/2 //

0 %%

Z/2⊕ Z/2

Z/2

Implying that η2 lifts to a non-trivial morphism S−8n−4 → P−8n−5 in dia-gram (4.3). Now, we wish to show that this lift is mapped to something nontrivial when composed with P−8n−5 → S−8n−5. Recall that P 1

0 ' S0 ∧ S1

and is the cofiber of S0 0→ S0. So, by periodicity we see that

0→ Z/8→ π−8n−3P−8n−5−8n−6 → Z/2→ 0

So, π−8n−3P−8n−5−8n−6 must be of order 16. We have the cofiber sequence

P−8n−5−8n−6 → P−8n−6 → P−8n−4, so we have the following commutative di-

agram

P−8n−6//

c(2)

P−8n−5

// S−8n−5

P−8n−4

P−8n−4// ∗

ΣP−8n−5−8n−6

ΣP−8n−6

Applying π−8n−2 to the diagram and using periodicity we see thatπ−8n−2(c(2)) = 0 since π−8n−2P−8n−4 and π−8n−3P−8n−6 are both of order 4.Now, a diagram chase similar to the one in lemma 4.7.1 and lemma 4.7.2 im-plies that S−8n−2 → P−8n−4 lifts to a non trivial morphismS−8n−2 → S−8n−5.

Since π−8n−2P−8n−5∼= Z/2⊕Z/2 we know that the non-trivial morphism

S−8n−2 → S−8n−5 has order two. This fact then in turn implies that themorphism is in the kernel of d1 and hence represents the generator of E8n+6,3

2 .This proves the lemma.

We note that the differentials are periodic in s, but that the periodicityincreases with both the page and t. Due to time constraints I have there-fore not computed any more differentials. To summarize the non-trivial d2

52


differentials we have drawn a section of the E2 page below with the all thenon-trivial differentials originating in the section drawn in. The fact thatthere are no other other non-trivial differentials originating in this sectioncan be seen in [MR93b].

t

−s−5 −4 −3 −2 −1 0 1

0

2

4

6

Z/20Z/20Z/20Z/2

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

0000000

0000000

Z/2Z/2Z/2Z/2Z/2Z/2Z/2

53

Chapter 5

The Complex Projectives

We will now turn our attention to the stunted complex projective spectra,which in many ways will be very similar to what we have studied so far.The goal is quite analogous to the ”real”-case: To define a complex Ma-howald filtration and root invariants. One could hope that the two caseswere completely identical: Alas this turns out to be false. Unfortunatelyhere is no complex version of The Kahn-Priddy Theorem (and hence neitherof Jones’ strengthening of it) and as we shall see the homotopy limit of thestunted complex projective spectra is far too large for our convenience. Bothof these issues significantly reduces the usefulness of the spectral sequencewhich will arise; yet still we press on. We will start by defining a stuntedcomplex projective spectra and then turn our attention to the complex Ma-howald filtration and root invariants. Finally we will construct a spectralsequence from the tower of stunted complex projective spectra.

5.1 Geometry

We now wish to construct stunted complex projective spectra. This willbe done similar to the real case, with the exception that there is no easyequivalent to the quadratic construction. We will not go into great detailas the definitions are almost identical to the real case. We start by notingthat we in definition 2.1.3 can replace R with C to get the definition of acomplex vector bundle. In the same manner we can define a morphism ofcomplex vector bundles and riemmanian metric. With this in mind we cannow give the following definition of the complex Thom spectrum, leaving itto the reader to fill out the details in the obvious way.

Definition 5.1.1. Given a complex (possibly virtual) vector bundle ξ we candefine the (complex) Thom spectra, Thom(ξ), in the exact same way as forthe real case.

54

CHAPTER 5 THE COMPLEX PROJECTIVES

Let C(γ)k denote the canonical complex line bundle

(L, v) ∈ CP k × Ck+1|v ∈ L

over CP k, then we have the following definition:

Definition 5.1.2. CP k+mm := Thom(m · C(γ)k), for all m ∈ Z and k ∈ N.

To see that this recovers the notion of stunted complex projective spaces,we note the following:

Lemma 5.1.3. CPm+km is homeomorphic to CPm+k/CPm−1 (as spaces) for

positive m.

Proof. We define a function f : S2k+1 × D2(m+1) → S2(k+m)+1, where wethink of S2i−1 ⊂ D2i ⊂ Ci, by f(x, y) = (y, (1− ‖y‖)x). Clearly

f(S2k+1 × S2m+1) = S2m+1 ⊂ S2(k+m)+1

Now, given λ ∈ S1 ⊂ C, then f(λ · x, λ · y) = λ · f(x, y), so we obtain aninduced function f : D(m · C(γ)k)→ CPm+k, such that

f(S(mC(γ)k)) = CPm−1

Observe now that since f : S2k+1 × intD2(m+1) → S2(k+m)+1 − S2m+1 is ahomeomorphism then so is g : D(mC(γ)k)−S(mC(γ)k)→ CPm+k−CPm−1,and hence the quotient map gives a homeomorphism from the Thom spaceto what we wanted.

We see again that for all n the inclusion CP k → CP k+1 induces theinclusions CP k+n

n → CPn+k+1n .

From [Rav84] we have the following result showing that (as in the realcase) the stunted complex projective spaces are (up to suspension) closedunder S-duals.

Lemma 5.1.4. The S-dual of CP k−1m is Σ2CP−m−1

−k .

We will again be needing the collapse map, and for convenience this timemerely define it as the dual to the inclusion.

Definition 5.1.5. We define the complex collapse map c : CP k−1m → CP k−1

m+1

to be the second desuspension of the dual to the inclusion

i : CPm−2−k → CPm−1

−k

We can easily see that CP km has the expected cell structure i.e. one cellin every even dimension between 2m and 2k (both included). So we get theobvious cofiber sequences from the inclusion

S2k+1 // CP kmi // CP k+1

m// S2k+2

55

Section 5.1 Geometry

which of course dualizes to the cofiber sequence

S2m // CP kmc // CP km+1

// S2m+1

where we can think of the leftmost morphism as inclusion of the bottom cell.As before we will use the inclusions to form CPm := hocolimkCP

km → CP k+1

m

Let CP−k denote the 2-completion of CP−k then we have the followingresult from [Rav84]:

Theorem 5.1.6. holimkCP−k ' S−22∨∏∞i=0 Σ−1CP0.

Note the pinch morphisms π : CP k−10 → S0, defined as for real projective

spectra, which dualize to compatible morphisms

C(ι) : S−2 → CP−1−k → CP−k

Since π is not trivial then neither is C(ι). So we obtain a non-trivial mor-phism C(ι) : S−2 → holimkCP−k. Further, since we see that

S0 → CP0π→ S0

is the identity, then so is

S−2 (C(ι)→ Σ−2D(CP0)→ S−2

Hence, S−2 (C(ι)→ holimmCP−m is a retract, so π∗(C(ι)) is injective. Clearlythe above holds as well if we 2-localize everything, more clearly then fromthe above we can see that C(ι) : S−1

2→ holimkCP−k must be the identity

on the wedge summand of S−22

in the identification above.

5.1.1 clASS

In an analogous to the real case we want to identify C(ι) in Bruner’s calcu-lations of the E2-page of the clASS converging to CP−k. In the same wayas in the real case we can see that C(ι) is detected by H∗(−,Z/2), so up tosome indeterminacy we know that C(ι) is detected in E0,−1

2 .

Further, we wish to ascertain that the sections printed in appendix Bare not misleading. Recall that H∗(CP−k;Z/2) ∼= Z/2[x, x−1]/x−k−1 with|x| = 2. Clearly sq1xj = 0 for degree reasons, so as an A(0) module this isisomorphic to an infinite sum of even (de)suspensions of Z/2. But due totheorem 1.5.2 this implies that when s− t < s then Es,t is trivial if s− t isodd and Z/2 if s− t is even and greater or equal to −2k. Thus we have nowseen that the sections printed in the appendix do not obscure data from us.

56


5.2 The Complex Mahowald Filtration and RootInvariant

As before let πt := πtS2, then in exactly the same way as before both wecan define the complex Mahowald filtration and root invariants. Followingthis we will then give some examples of the root invariants.

Definition 5.2.1. Define the complex Mahowald filtration, CM sπt ⊂ πt, tobe Ker [C(ι) : πt−2S

−22→ πt−2CP−s]

Clearly CM0πt = πt and by the above we have that⋂sCM

sπt = 0 soin the language of [Boa99] the filtration is exhaustive and Hausdorff.

Lemma 5.2.2. CM sπt is a complete filtration in the sense of [Boa99].

Proof. For t > 0 then CM sπt is finite so by The Mittag-Leffler Conditionthe right-derived limit of the tower M s+1πt →M sπt is trivial, so by [Boa99]Prop. 2.2 the filtration is complete. When t = 0 we see that M1π0 = 0 andhence we are done. This is seen by taking the long exact sequence for thecofiber sequence S−2 → CP−1 → CP0, giving us

. . . // π−1CP0// π−2S

−2 C(ι) // π−2CP−1// . . .

Clearly this implies that the kernel is trivial.

Definition 5.2.3. Given α ∈ CM sπt − CM s+1πt then we define the com-plex (Mahowald) root invariant of α, CR(α), as the set of all elementsβ ∈ πt−2S

−2s−22

such that

S−2s−22

$$St−2

α

##

β;;

CP−s−1

S−22

C(ι)::

commutes.

That this is well defined is inferred from the following cofiber sequence

S−2s → CP−s → CP−s+1

Note that for α ∈ CR(α) we do not have so nice a statement as for the realcase.

57

Section 5.2 The Complex Mahowald Filtration and Root Invariant

Lemma 5.2.4. α ∈ CR(α) if and only if C(ι) α : St−2 → CP−1 is non-trivial.

Following this we can now give some simple examples of the root invari-ants:

Lemma 5.2.5. Let η ∈ π1 and ν ∈ π3 be the standard Hopf fibrations. Thenν ∈ CR(η) and ν2 ∈ CR(η2).

Proof. By reading off from Bruner’s calculations we obtain the followingdiagrams proving our statements. Let x denote the generator of the Z2

summand of π−2CP−k whenever k ≥ 1. Then since C(ι) is detected by mod-2-cohomology we know that there is some k ∈ Z such that (2k+ 1)x = C(ι).

• ηx : S−1 → CP−2 is non-trivial since it could only be killed by hav-ing an exiting differential; but if it does have one of these then sincedr(2

iC(ι)η) = dr(2)C(ι)η + 2idr(C(ι)η) = 2idr(C(ι)η) must then benon-trivial, but 2iC(ι)η = 0 for high i, but then since C(ι) = (2k+1)xand η2 = 0, then ηC(ι) = ηx which is non trivial.

• Note that xη2 : S0 → CP−3 must be non-trivial since its representativein E2,2

2 can support no exiting differentials. The only differential whichcould hit it non-trivially would be d1 exiting E1,2, but since we are onthe E2-page this is not possible. By an argument similar to the oneabove above we see that the representative for νj in E1,−2 must surviveto the infinite page.

Thus we can now note the existence of the following commutative diagrams

S−4 // CP−2

S−6 // CP−3

S−1

0

44

ν

==

η // S−2 C(ι) //

C(ι)<<

CP−1 S0

0

44

ν2>>

η2 // S−2 C(ι) //

C(ι)<<

CP−2

And hence we are done.

One might wonder whether we could obtain a result similar to Jones’Kahn-Priddy theorem relating to C(ι) or even just a statement related toThe Kahn-Priddy Theorem. Unfortunately this is not the case. To see thisis so we will briefly recall some facts about the J homomorphism. A morecomplete account can be found in [Rav03].

Theorem 5.2.6. There exists a homomorphism J : πunstk SO → πkS2 withJ(πunst4k−1SO) ∼= Z(2)/8k.

58


Now, similarly to in the real case we wish to define a morphism C(τ) byextending the cofiber sequence relating to C(ι), so we define C(τ) by thecofiber sequence

S−22

C(ι) // CP−1c // CP0

C(τ) // S−1

Up to a sign this morphism is what Miller calls the S1-transfer and in [Mil82]he proves:

Theorem 5.2.7. For k > 0, C(τ) : π8k−2CP0 → π8k−1S02

does not hit thegenerator for the image of the J homomorphism,

implying that C(τ) is not surjective from some step onwards, denyingus any possibility of a Kahn-Priddy theorem for stunted complex projectivespectra.

5.3 The Spectral Sequence

One might be tempted to drive the analogy with the real projective spaceseven further and hope for the existence of a spectral sequence similar tothe Mahowalds stable EHP sequence. Such a convergent spectral sequenceexists. However, as lamented above unfortunately the target, of this spectralsequence, is much more complicated. We will continue to work mostly withthe 2-completed situation.

5.3.1 The set up

Let k : S−2s2→ CP−s be the 2-completion of inclusion of the bottom cell.

Then we have the following cofiber sequence

S−2s2

k // CP−sc // CP−s+1

j // S−2s+12

In the same way as earlier we define

As,t := πt−2s+1CP−s+1

Es,t := πt−2s+1S−2s+12

= πtS02

and we obtain (with some abuse of notation) maps:

j := πt−2s+1(j) : As,t → Es,t

k := πt−2s+1(k) : Es,t → As+1,t+1

i := πt−2s+1(c) : As,t → As−1,t−2

59

Section 5.3 The Spectral Sequence

This gives an unrolled exact couple in the sense of [Boa99]. And as beforewe can define the cycles, the boundaries and terms on the higher pages by

Zs,tr := k−1(Im [i(r−1) : As+r,t+2r−1 → As+1,t+1]

Bs,tr := j(Ker [ir−1 : As,t → As−r+1,t−2r+2])

Es,tr := Zs,tr /Bs,tr

The dr-differential dr : Es,tr → Es+r,t+2r−1r is induced by j (i(r−1))−1 k.

5.3.2 Convergence

We will now show the convergence of the spectral sequence. The strategywill be identical to that employed in section 4.2. Notice that by the sameargument as in section 4.2 colimsA

s,t = 0, so any convergence must betowards A∞. We will show that the right derived functors of the limit ofboth As,t and Es,t are trivial; in fact it will turn out that they are trivialwith almost the same argument.

Lemma 5.3.1. RA∞ = 0

Proof. Fix an odd integer n, then πnCPk is finite for all k and hence thefirst rightderived of the limit is trivial by the Mittag-Leffler condition.

Now fix n ∈ Z, n 6= −1 and take hk : π2nCPn−k → H2nCPn−k. Clearlyh0 is an isomorphism by the Hurewicz theorem. So by naturality we havethe following commutative diagram

π2nCPn−khk //

H2nCPn−k

∼=

π2nCPnh0∼=

// H2nCPn

Since we know that π−2nCP−n−k ∼= Z ⊕ Gk for some finite group Gk thenthere must be some dk ∈ Z such that hk = (·dk, 0), since rationalizationturns hk into an isomorphism dk 6= 0. Further by the following commutativediagram

π2nCPn−k−1hk+1 //

H2nCPn−k−1

∼=

π2nCPn−kh0 // H2nCPn−k

we can see that dk+1 = d′k+1dk for some d′k+1.

60


If we tensor the above case with Z2 we obtain the following diagram

π2nCPn−khk //

(·dk,0)

H2nCPn−k ⊗ Z2

∼=

π2nCPnh0∼=// H2nCPn ⊗ Z2

Now we wish to show that for all N ∈ N there exists k > N such that2|d′k. Assume for contradiction that this is not the case, then for some

N ∈ N for all n ≥ N we have that πnCP2n−k → πnCP2n−N restricted tothe Z2-summand hits a generator of the Z2-summand, implying that Z2 is a

subgroup of the limit. But since holimkCP−k ' S−22∨∏∞i=0 Σ−1CP0, then

this cannot hold, since this has finite homotopy groups in all even degreesexcept −2. Let Ker k be the kernel of hk ⊗ Z2 and Im k its image. Then forall k we have compatible short exact sequences

0 // Ker k // π2nCPn−k // Im k// 0

Now clearly Ker k is finite for all k so R limk Ker k = 0, which implies thatR limk π2nCPn−k ∼= R limk Im k, so let us compute R limk Im k. Note thatsince Im k = dkZ2, we have the following commutative diagram with exactrows

...

......

0 // d2Z2

//

Z2// Z2/d2Z2

// 0

0 // d1Z2 //

Z2// Z2/d1Z2

// 0

0 // Z2 // Z2

// 0 // 0

Now, taking the limit over the columns yields the following exact sequencefrom the

0 // 0 // Z2

ϕ // Z2// R limk Im k

// 0

but clearly ϕ is an isomorphism. Hence R limk Im k is trivial implying thatRA∞ = 0.

Now take π−2CP−1−k. By the same argument as above we see thatR limk π−2CP−1−k ∼= R lim Im k where

Im k = Im [hk : π−2CP−1−k → H−2CP−1−k ⊗ Z2]

61

Section 5.3 The Spectral Sequence

We can also define the dk’s as above. Now, it is true or false that N ∈ Nthere is k > N such that 2|d′k. If true, then a similar argument to the one

above will show that R limk π−2CP−1−k = 0. If false, then there exists someN ∈ N such that Im k → ImN is an isomorphism for all k ≥ N , but then bythe Mittag-Leffler condition we have shown that R lim Im k = 0 and hencewe are done.

We will now relate the argument above to the issue of Es,tr in order toarrive at the following statement:

Lemma 5.3.2. RE∞ = 0

Proof. This is clear whenever t 6= 0 since then Es,t is finite and hence so isall of Es,tr and therefore so is REs,t∞ .

Thus, take t = 0. Since Es,0r allows for no non-trivial entering differen-tials, then Es,0r = Zs,0r . Hence this depends on the following diagram:

π−2sCP−s−r+1

i(r−1)

π−2sS

−2s2

k // π−2sCP−s // π−2sCP−s+1

in the sense that Zs,0r is the preimage under k of the image of i(r−1); butsince k is an isomorphism in the diagram above, we can exactly use theargument above.

We are now ready to conclude that our spectral sequence does in factconverge to the homotopy limit.

Theorem 5.3.3. There exists a spectral sequence with Es,t1 := πt−2s+1S−2s+12

and differential dr : Es,tr → Es+r,t+2r−1r which converges to π∗holimkCP−k.

Proof. Of course this is the spectral sequence that we have been designingthroughout this section. By an argument analogous to theorem 4.2.4 we canshow that this spectral sequence do in fact converge.

Now, as for the real case we might ask ourselves if this spectral sequencerelates to the complex root invariants.

Proposition 5.3.4. Given α : St−2s → S−22

(non-trivial) with

α ∈ CM sπt−2(s−1) − CM s+1πt−2(s−1)

then if β ∈ CR(α), then β represents a non trivial element on the E∞-pageand this element represents C(ι) α.

62


Proof. By [Boa99] we know that the filtration of the target is

F s,t := Ker [πt−2sholimkCP−k → πt−2sCP−s] ≥ CM sπt−2(s−1)

Recall that C(ι) : S−22→ holimkCP−k is inclusion of the sphere summand

and hence injective on homotopy groups. Thus our assumptions imply thatC(ι)α is represented in Es,t∞ . And by similar arguments as for the real casewe are done.

Note that for degree reasons we know that Es,t∞ = 0 whenever t−2s < −2,but without something similar to The Kahn-Priddy Theorem we cannotobtain the same wedge shape as for the other spectral sequence.

63

Outlook

In this chapter we will refer the reader to some results and articles that areclosely related to the subject of the thesis. The hope is to motivate furtherstudy into the topics presented in the thesis.

The stunted real projective spectra

Given a finite spectrum X, it is well known that holimkP−k∧X ' Σ−1X2. Ingeneral nothing is known unless X is finite. In [DM84] Davis and Mahowaldmade the following conjecture:

Conjecture 5.3.5. holimkP−k ∧BP 〈n〉 '∏k∈Z Σ2k−1BP 〈n− 1〉

where BP 〈n〉 is the Brown-Peterson spectrum associated to 2 and BP 〈n〉its 2-completion. In the same article this was shown for n = 1 and in[DJK+86] it was proven for n = 2, but for n ≥ 3 the conjecture is still open.

Further, for each spectrum X, P−k∧X again leads to spectral sequences,as do various localizations, all of which might converge and be of more or lessinterest, but perhaps one could find some worthy of further study. Speakingof spectral sequences, the spectral sequence presented in chapter 4 is wellstudied, and, as one might expect, quite a bit more is known about it thanpresented here, e.g., several more of the d2 differentials.

The root invariant

As concerns the root invariant, several articles exist presenting calculationsor broadening of the concept. In [MR93a] it is shown that αi ∈ R(2i)for i > 1 where αi is the lowest dimensional element of π∗S

0 having Adamsfiltration i. This is a rather elegant result as it implies that the root invariantof the element represented by Ei,i2 in the classical Adams Spectral Sequencefor the sphere spectrum has its root invariant represented by the first ”dot”to the right of it.

The proof requires two variations on the root invariant. Recall that theoriginal root invariant came from the cofiber sequence S−k → P−k → P−k+1.We will not give a complete definition of the variations, but simply note

64

OUTLOOK

the relevant cofiber sequence. The first variant comes form the cofibrationΣ−2k−1M(2) → P−2k−1 → P1−2k and is called the modified root invariantR′. Given a ring spectrum E we obtain new cofiber sequences by smashingwith E. The unit map S0 → E allows us to define the E-root invariantRE , and the modified E-root invariant R′E (Note that RE and R′E are notdefined for all α ∈ π∗S0). The proof now comes from calculations of R′bo,where bo is connective K-theory, and then linking the various concepts.

Miller [Mil90] offers a different approach to proving αi ∈ R(2i) by cal-culating bo−1P−k and J−1P−k, which will ensure that the dimensions of theroot invariants of R(2i) fit with αi and from here an argument on the Adamsfiltration completes the proof.

As stated earlier our interest in root invariants comes from the hopethat they will allow us to find (up to some indeterminacy) high dimensionalelements of π∗S

0. This poses a set of obvious questions of the form: If αhas some property, then what do I know about the elements of R(α)? Theanswer to one such question can be found in [MS88]. A finite 2-completedCW-spectrum X has a νn self map if there exists a morphism ν : ΣkX → Xsuch that K(m)∗(ν) = 0 if m < n and an isomorphism when m = n, whereK(m) is the m’th Morava K-theory. If X has a νn self map then [X,−]∗ isa module over Z[ν]. Given α ∈ π∗(X) (the stable cohomotopy groups of X)then we can define R(α) to be the morphisms making the following diagramcommute

Σj−1Xα //

R(α)

S−1

ι

S−k // P−k

Now, Mahowald and Shick showed that if α is ν-periodic then R(α) is ν-torsion.

Primes not equal to 2

A phrase heard from time to time when talking about stable homotopytheory is: ”There are only two primes, 2 and the odd ones”. For the entiretyof this thesis we have been focusing on the case of p = 2, but most of ourconstructions are equally valid for for odd primes. Similar to the stuntedreal projective spectra one can construct stunted lens spectra for odd primesin analogous ways, by Thom spectra and something similar to the quadraticconstruction. In this situation both The Kahn-Priddy Theorem and Jones’strengthening still hold true, with a completely analogous proof. Further-more, we have Gunawardena’s Theorem stating that ι : S−1 → homlimkP−kis the p-completion when p 6= 2. It was this fact that lead to the study of

65

Primes not equal to 2

the complex projective spectra in the hope that the generalization wouldextend even further than merely to the odd primes.

66

Appendix A

E2 for clASS of P−s

Here are Bruner’s calculations of the E2 page of the classical Adams Spec-tral Sequence for P−s, which have been taken from [Bru14]. Note that thebottom left corner on each page has the coordinates (0,−k) for the spectralsequence converging to P−k, which is marked top left as

P_-k/A from s,n=0,-k

Included are P−1 to P−9. I will remind the reader that the vertical axis is sand the horizontal axis t− s. The thick vertical lines denote multiplicationby h0 (which detects 2 in the spectral sequence for the sphere spectrum),the diagonals multiplication by h1 (which detects η ∈ π1S

0) and the dashedlines multiplication by h2 (which detects ν ∈ π3S

0).

67

P_-1/A from s,n=0,-1

E2 FOR clASS OF P−s

68


APPENDIX A

69



70


APPENDIX A

71



72


APPENDIX A

73



74


APPENDIX A

75



76

Appendix B

E2 for clASS of CP−s

Here are Bruner’s calculations of the E2 page of the classical Adams SpectralSequence for CP−s, which I have recieved from my advisor Professor MarkusSzymik. Note that the bottom left corner on each page has the coordinates(0,−2k) for the spectral sequence converging to CP−k, which is marked topleft as

CP_-k/A from s,n=0,-2k

Included are CP−1 to CP−9. I will remind the reader that the verticalaxis is s and the horizontal axis is t − s. The thick vertical lines denotesmultiplication by h0 (which detects 2 in the spectral sequence for the spherespectrum), the diagonals multiplication by h1 (which detects η ∈ π1S

0) andthe dashed lines multiplication by h2 (which detects ν ∈ π3S

0).

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CP_-1/A from s,n=0,-2

E2 FOR clASS OF CP−s

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APPENDIX B

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APPENDIX B

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APPENDIX B

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APPENDIX B

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Appendix C

E2 for clASS of S0 and M(2)

Here are Bruner’s calculations of the E2 page of the classical Adams SpectralSequence for S0 and the mod-2 Moore spectrum M(2), which have beentaken from [Bru14]. Note that the bottom left corner on each page hasthe coordinates (0, 0) for both of the spectral sequences. The next pagewill be E2 for S0 and the following page E2 for M(2). I will remind thereader that the vertical axis is s and the horizontal axis t − s. The thickvertical lines denote multiplication by h0 (which detects 2 in the spectralsequence for the sphere spectrum), the diagonals multiplication by h1 (whichdetects η ∈ π1S

0) and the dashed lines multiplication by h2 (which detectsν ∈ π3S

0).The reason for including E2 for S0 is twofold. Firstly, we have throughout

the thesis several places needed the stable homotopy groups of the sphere.Secondly, an interpretation of The Kahn-Priddy Theorem is, that the pat-tern found here should slowly emerge starting in t− s = −1 for P−k when kgrows. In fact it was shown in [LDMA80] that ι induces an isomorphism

E2(S−1) ∼= limkE2(P−k)

Where E2(X) is the E2-page of the classical Adams Spectral Sequence con-verting to X2.

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E2 FOR clASS OF S0 AND M(2)

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APPENDIX C

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[KP72] Daniel S. Kahn and Stewart B. Priddy, Applications of the trans-fer to stable homotopy theory, Bull. Amer. Math. Soc 78 (1972),no. 1972, 135–146.

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[Lin80] Wen-Hsiung Lin, On conjectures of Mahowald, Segal and Sulli-van, Mathematical Proceedings of the Cambridge PhilosophicalSociety, vol. 87, Cambridge Univ Press, 1980, pp. 449–458.

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The Kahn-Priddy Theorem202014/JensJako… · Abstract In this thesis we will prove The Kahn-Priddy Theorem in stable homo-topy theory. The theorem states that there exists a morphism

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