EULER CHARACTERISTICS OF p-LOCAL COMPACT GROUPS€¦ · 2. The Euler Characteristic of Finite Categories 3 3. p-local compact groups 5 3.1. Discrete p-toral groups and fusion systems

EULER CHARACTERISTICS OF p-LOCAL COMPACT GROUPS

ANDREW GEORGE BATALLAS, ALEX GONZALEZ, MITCHELL MESSMORE,AND ANDREW SMITH

Abstract. In 2008 Tom Leinster defined the Euler characteristic of a finitecategory in a manner which is compatible with the Euler characteristics oftheir classifying spaces [8]. We present a generalization of this definition to theEuler characterisitic of certain infinite categories, specifically those of linkingsystems of the p-local compact groups of Broto, Levi, and Oliver [2]. This hasapplications to the fusion theory and p-local homotopy theory of compact Liegroups.

The authors were partially supported by the Kansas State University Summer UndergraduateMathematics Research program and NSF grant DMS-1262877.

1

2 A. BATALLAS, A. GONZALEZ, M. MESSMORE, AND A. SMITH

Contents

1. Introduction 22. The Euler Characteristic of Finite Categories 33. p-local compact groups 53.1. Discrete p-toral groups and fusion systems 53.2. Linking systems and transporter systems 73.3. Unstable Adams operations on p-local compact groups 84. The Euler characteristic of p-local compact groups 94.1. A Finite Retract of a p-Local Compact Group 94.2. The Euler Characteristic as a Limit 114.3. Fusion and Transporter Categories 135. Future Work 13Appendix A. Related Work 14A.1. Grothendieck Topologies 15A.2. Sheaves on a Site 16A.3. Final Thoughts 18References 18

1. Introduction

Given a group G, there are various notions of studying it by isolating its proper-ties at a prime p. For example, suppose we were studying the fusion properties ofa group – its subgroups, and the data of which group elements conjugate di↵erentsubgroups onto each other, or “fuse” them. We could localize at a prime p by onlyconsidering fusion among the p-subgroups of G. Another notion of localization atp would be to consider the mod-p group cohomology of G. Both of these kinds oflocal data at p are determined, in good circumstances, by homotopy type of thep-completion BG^

p

of the classifying space of G [2].Indeed, if G is finite, we can collect all of the information of the p-local fusion in

the group into an object known as a “p-local finite group.” This object, discussedin [4], consists of a Sylow S of G and two categories F and L, satisfying certainaxioms given in that section. Then we can find BG^

p

, up to homotopy equivalence,as the p-completed nerve of L. For this reason, we consider this p-completed nerveto be the “classifying space” of the p-local finite group.

In [8] Tom Leinster developed an idea of an Euler characteristic for finite cat-egories, discussed in section 2. This notion of Euler characteristic is based in anotion of Mobius inversion generalizing that developed by Rota for partially or-dered sets [9]. In particular, in good conditions it agrees with the traditional Eulercharacteristic of the nerve of the category when that exists. Thus the Euler char-acteristic �pLq is a natural object of study. In [6], it was shown how to computeEuler characteristics for various associated categories arising from finite groups.

But what about when our group G is not finite? A natural generalization wouldbe compact Lie groups, because they share many of the properties of finite groups– indeed we may see finite groups with the discrete topology as exactly the 0-dimensional compact Lie groups. In [2], the notion of a p-local finite group wasgeneralized from finite groups to “p-local compact groups”, and in [3] it was shownthat any p-completed classifying space of a finite loop space is the classifying space

EULER CHARACTERISTICS OF p-LOCAL COMPACT GROUPS 3

of some p-local compact group. These p-local compact groups, discussed at lengthin section 3, are still triples of a p-group and two associated categories. However,neither the p-group nor the categories need be finite, posing an obstacle to givingthe same treatment to their Euler characteristics. It is the goal of this paper topresent a definition of an Euler characteristic for a p-local compact group, and arguethat it has some nice properties that we would expect of an Euler characteristic.In particular, we will present a retract of the category with finitely many objects,and then show that if we approximate this by finite categories, the limit alwaysconverges to the same value in good circumstances.

The paper will be organized as follows. Section 2 will introduce the concept ofthe Euler characteristic of a finite category and many useful formulas for computingit. Section 3 will include background information on p-compact local groups. Insections 4 and 5 the Euler characteristic of F and L will be defined and discussed.The paper will conclude with open problems that still need to be addressed andwork related to p-local compact groups. Finally, the appendix will present a mostlyunrelated analysis of a geometric interpretation of saturated fusion systems.

2. The Euler Characteristic of Finite Categories

In this section we shall briefly introduce the Euler characteristic of a finite cat-egory. See [8] for more detail. For the rest of this section take C to be a finitecategory.

Definition 2.1. [8, Definition 1.1] Let RpCq represent the Q-algebra of functionsObpCq ˆ ObpCq Ñ Q, with pointwise addition and scalar multiplication, the Kro-necker � as unit, and multiplication defined by

p✓�qpa, cq “

ÿ

bPObpCq✓pa, bq�pb, cq

for ✓,� P RpCq and a, c P pOb Cq.The zeta function ⇣ “ ⇣C P RpCq is defined by ⇣pa, bq “ |MorCpa, bq|. If

⇣ is invertible in RpCq then C is said to have Mobius inversion; its inverseµ “ µC “ ⇣´1 is the Mobius function of C.Remark 2.2. If the objects of C are totally ordered, a1, a2, ¨ ¨ ¨ , a

n

P ObpCq, then itmay be helpful to picture ⇣ and µ as n ˆ n matrices

Z “

¨

˚⇣pa1, a1q ¨ ¨ ¨ ⇣pa1, anq

.... . .

⇣pan

, a1q ¨ ¨ ¨ ⇣pan

, an

q

˛

‹‚

M “ Z´1“

¨

˚µpa1, a1q ¨ ¨ ¨ µpa1, anq

.... . .

µpan

, a1q ¨ ¨ ¨ µpan

, an

q

˛

‹‚

Example 2.3. Let G be a finite group. Interpret G as a category G, i.e. ObpGq “ ˚,MorGp˚, ˚q “ G, and composition is simply the group multiplication of G. Then⇣p˚, ˚q “ |G| and so µp˚, ˚q “

1|G| .


Definition 2.4. [8, Definition 1.10] Let C be a finite category. A weighting on Cis a function k‚ : ObpCq Ñ Q such that for all a P ObpCq,

ÿ

bPObpCq⇣pa, bqkb “ 1.

A coweighting k‚ on C is a weighting on Cop.

Remark 2.5. In the perspective of remark 2.2, the weighting can be thought of asthe solution to the linear equation Zk‚

“

~1, where

k‚“

¨

˚ka1

...kan

˛

‹‚

Example 2.6. (a) Let W be the category pictured below, where g2 “ h2“ 1

A

,gh “ hg, ↵h “ ↵, ↵g “ �, and �g “ �. Then the unique weighting on W iskA “ ´

14 , k

B

“ 1.

A B

1A

g

h

gh

1B

↵

�

(b) Let Y be the category pictured below, where ˝ '1 “ �1 and ˝ '2 “ �2.Then the unique weighting on Y is kA “ ´2, kB “ 0, and kC “ 1.

A

B

C

1A

1B

1C

'1

'2

�1

�3

�2

The definition of the Euler characteristic relies on the following lemma, which isproved by applying a partition of unity on each side:

Lemma 2.7. [8, lemma 2.1] Let C be a finite category, k‚a weighting on C, and

k‚ a coweighting on C. Then

∞aPObpCq k

a

“

∞aPObpCq ka.


Definition 2.8. [8, Definition 2.2] A finite category C has Euler characteristicif it admits both a weighting and coweighting. Its Euler characteristic is then

�pCq “

ÿ

aPObpCqka “

ÿ

aPObpCqka

Remark 2.9. It is possible for solutions to Zk‚“

~1 and k‚Z “

~1 to exist evenwhen Z is singular. However, if Z is invertible, we have another characterizationof the Euler characteristic: it is the sum of the entries of M “ Z´1. Hence Mobiusinversion always implies the existence of the Euler characteristic.

Example 2.10. (a) Recall the category W from example 2.6. Using the weightingkA “ ´

14 , k

B

“ 1 to calculate �pWq “

34

(b) Similarly we can use the weighting kA “ ´2, kB “ 0, and kC “ 1 of Y givenin example 2.6 to calculate �pYq “ ´1.

(c) From example 2.3 we see that the category G which arises from a finite groupG will have weighting k˚

“

1|G| and consequently �pGq “

1|G| .

The following theorems verify the usefulness and validity of definition 2.8. Thefirst establishes the Euler characteristic as a categorical invariant, while the secondrelates the Euler characteristic of a category to that of its classifying space. Bothallow us to think of the Euler characteristic as a homotopy invariant, justifying itsname.

Theorem 2.11. [8, Proposition 2.4] Let C and D be finite categories. If there is a

pair of adjoint functors between C and D, and both Euler characteristics exist, then

�pCq “ �pDq. Moreover, if C is equivalent to D then C has Euler characteristic if

and only if D does.

Theorem 2.12. [8, proposition 2.11] Let C be a finite skeletal category and BC be

the nerve of C. Then �pCq “ �pBCq when both exist.

3. p-local compact groups

3.1. Discrete p-toral groups and fusion systems.

Definition 3.1. Consider Z{p8, the colimit of the cyclic groups Z{pn along withembeddings between them via multiplication by p.

A discrete p-torus is a group T isomorphic to a finite direct product of copiesof Z{p8.

A discrete p-toral group P is an extension of a finite p-group ⇡ by a discretep-torus T . For such a group, we call T – pZ{p8

q

r the maximal torus of P , anddefine the rank of P as r.

Discrete p-toral groups were characterized in [2, Proposition 1.2] as those groupssatisfying the descending chain condition and such that every finitely generatedsubgroup is a finite p-group (that is, they are the locally finite Artinian p-groups).In particular note that a discrete p-toral group is a finite p-group if and only if itsmaximal torus is trivial.

Discrete p-toral subgroups of infinite groups play an important role analogousthe p-subgroups of finite groups. For an infinite group G, we say that G has Sylow

p-subgroups if G contains a discrete p-toral group S such that any finite p-subgroupof G is G-conjugate to a subgroup of S. If this is the case, we may consider all of


the p-local information of G to be contained in the subgroups of S, along with thedata of the G-conjugations between them.

Formally, for a group G and subgroups P, P 1§ G, define the transporter

subgroup NG

pP, P 1q to be the subgroup of elements of G which send P into P 1 by

conjugation. That is,

NG

pP, P 1q “ tg P G | gPg´1

§ P 1u.

Then define HomG

pP, P 1q “ N

G

pP, P 1q{C

G

pP q to be the set of homomorphismsinduced by the transporter subgroup.

In the case where P 1“ P , we also refer to this as Aut

G

pP q. Note that AutP

pP q “

P {ZpP q “ InnpP q.We are now ready to define fusion systems over discrete p-toral groups. Note that

fusion systems were originally developed over finite p-groups, cf. [4], and were laterextended to discrete p-toral groups in [2]. We will not discuss the finite construction,since it is merely the special case in this construction where the p-toral group hasrank 0.

Definition 3.2. A fusion system F over a discrete p-toral group S is a categorywhose objects are the subgroups of S and whose morphism sets HomF pP, P 1

q satisfythe following conditions:

(i) HomS

pP, P 1q Ñ HomF pP, P 1

q Ñ InjpP, P 1q for all P, P 1

§ S.(That is, the morphisms in F are injective group homomorphisms, and

include all conjugations in S).(ii) Every morphism in F factors as an isomorphism followed by an inclusion.

Remark 3.3. Given a fusion system F over a discrete p-toral group S, we will oftenrefer to the maximal torus and rank of F , which will refer to those of S.

Example 3.4. Let G be a compact Lie group. Then G has Sylow p-subgroups forany prime p; choose such a discrete p-toral subgroup S. Define the category F

S

pGq

with all the subgroups of S as objects, and

MorFSpGqpP, P 1q “ Hom

G

pP, P 1q.

We see that NS

pP, P 1q § N

G

pP, P 1q, so Hom

S

pP, P 1q § Hom

G

pP, P 1q § InjpP, P 1

q

as desired. Moreover, every map f : P Ñ Q factors through its image as P Ñ

fpP q Ñ Q, where the first map is an isomorphism by injectivity, and the second isthe inclusion.

Thus FS

pGq is a fusion system over S. Indeed, it is exactly the categories ofthis form which motivate the definition of a fusion system. Yet F

S

pGq has evennicer properties, which are not captured by this definition alone. Namely, it is asaturated fusion system, which we must now define.

First let us establish some useful terminology.

Definition 3.5. Let F be a fusion system over a discrete p-toral group S.

(1) Two subgroups P, P 1 are called F-conjugate if there is an isomorphismbetween them in F . For a subgroup P § S, we denote

PF“ tP 1

§ S | P 1 is F-conjugate to P u.


(2) For a discrete p-toral group P , the order of P is the ordered pair |P | “

prkP, |P {TP

|q, where TP

is the maximal torus of P . Thus, given two dis-crete p-toral groups P and Q we say that |P | § |Q| if either rkP † rkQ,or rkP “ rkQ and |P {T

P

| § |Q{TQ

|.

Definition 3.6. Let F be a fusion system over a discrete p-toral group S. Asubgroup P § S is called fully F-normalized (resp. fully F-centralized) if|N

S

pP 1q| § |N

S

pP q| (resp. |CS

pP 1q| § |C

S

pP q|) for all P 1P PF .

The fusion system F is called saturated if the following conditions hold:

(I) For each P § S which is fully F-normalized, P is fully F-centralized, andthe outer automorphism group OutF pP q “ AutF pP q{ InnpP q is finite andcontains Out

S

pP q “ AutS

pP q{ InnpP q as a p-Sylow subgroup.(II) If P § S and f P HomF pP, Sq is such that fpP q is fully F-centralized, then

there exists rf P HomF pNf

, Sq such that f “

rf |

P

, where

Nf

“ tg P NS

pP q | f ˝ cg

˝ f´1P Aut

S

pfpP qqu.

(III) If P1 § P2 § P3 § . . . is an increasing sequence of subgroups of S, with

P “

8§

n“1

Pn

,

and if f P HompP, Sq is any homomorphism such that f |

Pn P HomF pPn

, Sq

for all n, then f P HomF pP, Sq.

Note that all the automorphism groups in a saturated fusion system are Artinianand locally finite. The condition in axiom (I) of OutF pP q being finite is in factredundant, as was pointed out in [2, Lemmas 2.3 and 2.5], where the authors showthat the set RepF pP,Qq “ InnpQqzHomF pP,Qq is finite for all P,Q P ObpFq.

Given a discrete p-toral group S and a subgroup P § S, we say that P is centricin S, or S-centric, if C

S

pP q “ ZpP q. We next define F-centric and F-radicalsubgroups.

Definition 3.7. Let F be a saturated fusion system over a discrete p-toral group.A subgroup P § S is called F-centric if all the elements of PF are centric in S:

CS

pP 1q “ ZpP 1

q for all P 1P PF .

A subgroup P § S is called F-radical if OutF pP q contains no nontrivial normalp-subgroup:

Op

pOutF pP qq “ t1u.

Clearly, F-centric subgroups are fully F-centralized, and conversely, if P is fullyF-centralized and S-centric, then it is F-centric.

3.2. Linking systems and transporter systems. Linking systems are the thirdand last ingredient needed to form a p-local compact group.

Definition 3.8. Let F be a saturated fusion system over a discrete p-toral groupS. A centric linking system associated to F is a category L whose objects arethe F-centric subgroups of S, together with a functor

⇢ : L ›Ñ Fc

and “distinguished” monomorphisms �P

: P Ñ AutLpP q for each F-centric sub-group P § S, which satisfy the following conditions.


(A) ⇢ is the identity on objects and surjective on morphisms. More precisely, foreach pair of objects P, P 1

P L, ZpP q acts freely on MorLpP, P 1q by composition

(upon identifying ZpP q with �P

pZpP qq § AutLpP q), and ⇢ induces a bijection

MorLpP, P 1q{ZpP q

–›Ñ HomF pP, P 1

q.

(B) For each F-centric subgroup P § S and each g P P , ⇢ sends �P

pgq P AutLpP q

to cg

P AutF pP q.(C) For each ' P MorLpP, P 1

q and each g P P , the following square commutes inL:

P' //

�P pgq✏✏

P 1

�P 1 phq✏✏

P'

// P 1,

where h “ ⇢p'qpgq.

A p-local compact group is a triple G “ pS,F ,Lq, where S is a discrete p-toral group, F is a saturated fusion system over S, and L is a centric linking systemassociated to F . The classifying space of G is the p-completed nerve

BG def

“ |L|

^p

.

Given a p-local compact group G, the subgroup T § S will be called the maximaltorus of G, and the rank of G will then be the rank of the discrete p-toral groupS.

We will in general denote a p-local compact group just by G, assuming thatS is its Sylow p-subgroup, F is the corresponding fusion system, and L is thecorresponding linking system.

As expected, the classifying space of a p-local compact group behaves “nicely”,meaning that BG “ |L|

^p

is a p-complete space (in the sense of [1]) whose funda-mental group is a finite p-group, as proved in Proposition 4.4 [2].

Finally, we note that the centric linking system over F is a special case of atransporter system over F , as defined in [2]

3.3. Unstable Adams operations on p-local compact groups. To concludethis section, we introduce unstable Adams operations for p-local compact groupsand their main properties. Essentially, we summarize the work from [7] in order togive the proper definition of such operations and the main properties that we willuse in later sections.

Let pS,Fq be a saturated fusion system over a discrete p-toral group, and let✓ : S Ñ S be a fusion preserving automorphism (that is, for each f P MorpFq,the composition ✓ ˝ f ˝ ✓´1

P MorpFq). The automorphism ✓ naturally inducesa functor on F , which we denote by ✓˚, by setting ✓˚pP q “ ✓pP q on objects and✓˚pfq “ ✓ ˝ f ˝ ✓´1 on morphisms.

Definition 3.9. Let G “ pS,F ,Lq be a p-local compact group and let ⇣ be a p-adicunit. An unstable Adams operation on G of degree ⇣ is a pair p , q, where is a fusion preserving automorphism of S, is an automorphism of L, and suchthe following is satisfied:

(i) restricts to the ⇣ power map on T and induces the identity on S{T ;(ii) for any P P ObpLq, pP q “ pP q;


(iii) ⇢ ˝ “ ˚ ˝ ⇢, where ⇢ : L Ñ F is the projection functor; and(iv) for each P,Q P ObpLq and all g P N

S

pP,Qq, p�P,Q

pgqq “ � pP q, pQqp pgqq.

In particular, is an isotypical automorphism of L in the sense of [2].

For a p-local compact group G, let AdpGq be the group of unstable Adams oper-ations on G, with group operation the composition and the indentity functor as itsunit. Also, for a positive integer m, let �

m

ppq § pZ^p

q

ˆ denote the subgroup of allp-adic units ⇣ of the form 1 ` pmZ^

p

.Next, we state the existence of unstable Adams operations for all p-local compact

groups. The following result corresponds to the second part of Theorem 4.1 [7].

Theorem 3.10. Let G be a p-local compact group. Then, for any su�ciently large

positive integer m there exists a group homomorphism

(1) ↵ : �m

ppq ›Ñ AdpGq

such that, for each ⇣ P �m

ppq, ↵p⇣q “ p , q has degree ⇣.

There is an important property of unstable Adams operations which we will userepeatedly in the forthcoming sections. This was stated as Corollary 4.2 in [7].

Proposition 3.11. Let G be a p-local compact group, and let P Ñ ObpLq andM Ñ MorpLq be finite subsets. Then, for any su�ciently large positive integer m,and for each ⇣ P �

m

ppq, the group homomorphism ↵ from (1) satisfies ↵p⇣qpP q “ Pand ↵p⇣qp'q “ ' for all P P P and all ' P M.

Remark 3.12. Let p , q be an unstable Adams operation on a p-local compactgroup G. By point (iv) in 3.9, ˝ �

S

“ � ˝ : S Ñ AutLpSq, and hence theautomorphism is completely determined by . Thus, for the rest of this paperwe will make no mention of (unless necessary) and refer to the unstable Adamsoperation p , q just by .

4. The Euler characteristic of p-local compact groups

Given a p-local compact group G “ pS,F ,Lq the Euler characteristic of G shouldbe a topological invariant of the classifying space of L. As |L| is an infinite loopspace and almost never has an Euler characteristic, we seek to define the Eulercharacteristic of G as the categorical Euler characteristic of L. But L is not finite inmost cases, so its Euler characteristic is not given by definition 2.8. Our approachconsists of approximating L with a family of finite subcategories C

k

Ä L thatpreserve the structure of L.4.1. A Finite Retract of a p-Local Compact Group. There is, of course, a bigdi↵erence between working with finite p-groups and with discrete p-toral groups: thenumber of conjugacy classes of subgroups. Fortunately, in [2] the authors produceda way of getting rid of infinitely many conjugacy classes while keeping the structureof a given fusion system.

This construction will be rather important in this paper, and we reproduce it herefor the sake of a better reading. Let then pS,Fq be a saturated fusion system overa discrete p-toral group, and let T be the maximal torus of F and W “ AutF pT q.

Definition 4.1. Let F be a saturated fusion system over a discrete p-toral groupS, and let e denote the exponent of S{T ,

e “ exppS{T q “ mintpk | xp

k

P T for all x P Su.


(i) For each P § T , let

IpP q “ tt P T | !ptq “ t for all ! P W such that !|P “ idP

u,

and let IpP q0 denote its maximal torus.(ii) For each P § S, set P res

“ txp

e

| x P P u § T , and let

P ‚“ P ¨ IpP res

q0 “ txt|x P P, t P IpP resq0u.

(iii) Set H‚“ tP ‚

|P P Fu and let F‚ be the full subcategory of F with object setObpF‚

q “ H‚.

Indeed, as pointed out in [2], we could extend this definition. By replacing P rms

with P rm`ks, we get a new object P ‚k“ P ¨ IpP rm`ks

q. In this way we may definea family of “bullet subcategories” F‚

“ F‚0,F‚1,F‚2, . . . .The following is a summary of section §3 in [2].

Proposition 4.2. Let F be a saturated fusion system over a discrete p-toral groupS. Then,

(i) the set H‚ contains finitely many S-conjugacy classes of subgroups of S; and(ii) every morphism pf : P Ñ Qq P MorpFq extends uniquely to a morphism

f‚ : P ‚Ñ Q‚.

This makes p q

‚ : F Ñ F‚ into a functor. This functor is an idempotent functor(pP ‚

q

‚“ P ‚), carries inclusions to inclusions (P ‚

§ Q‚ whenever P § Q), and isleft adjoint to the inclusion functor F‚

Ñ F .In particular, this last fact implies that the classifying space of F‚ is a deforma-

tion retract of the classifying space of F , and thus homotopy equivalent. Similarly,we will extend this retraction to a retraction L Ñ L‚. Hence in any reasonabledefinition of an Euler characteristic, �pLq should be the same as �pL‚

q. For ourpurposes, we will take this as a defining property of �pLq.

Definition 4.3. Let F be a saturated fusion system over a discrete p-toral group S,and let Fc‚

Ñ Fc be the full subcategory whose objects are the F-centric subgroupscontained in H‚

pFq. Now let L be a centric linking system associated to F . DefineL‚

Ñ L to be the full subcategory with ObpL‚q “ ObpFc‚

q.

Our retraction F Ñ F‚ descends to a retraction L Ñ L‚ as desired, see [2,4.2(a)]. Furthermore, since L‚ has as objects a subset of the objects of F‚, andtwo objects are isomorphic in L‚ whenever they are isomorphic in F‚, L‚ also hasfinitely many conjugacy classes of objects. Thus passing to the skeleton, rL‚

s hasfinitely many objects. Yet skeletons are equivalent categories by definition, so theclassifying space of rL‚

s is homotopy equivalent to that of L‚ and L. Thus it willsu�ce to define an Euler characteristic on this retract instead.

But even though rL‚s has finitely many objects, it still may have infinitely many

morphisms between them, and so we still cannot apply Leinster’s definition of Eulercharacteristic to it. We will further approximate it with an increasing sequence offinite subcategories.

Definition 4.4. Let ⇣ be a p-adic unit and be an unstable Adams operation ofdegree ⇣. Define

k

“ p

k

and the subcategory Ck

Ñ rL‚s to be the fixed points:

ObpCk

q “ tP P ObprL‚sq |

k

pP q “ P u and

MorCkpP,Qq “ t' P MorrL‚spP,Qq | k

p'q “ 'u


By theorem 3.10, the construction of Ck

is always possible. Moreover, Ck

isalways a finite category [5, Theorem A].

4.2. The Euler Characteristic as a Limit. Since each Ck

is finite, we may useDefinition2.8 to compute their Euler characteristics. Now rL‚

s is a union (i.e. acolimit) of these C

k

, and indeed, [5, Theorem A] tell us that the classifying spaceof L is also the homotopy colimit of the classifying spaces of the C

k

. Thus it seemsreasonable to define its Euler characteristic as follows:

Definition 4.5. Let G “ pS,F ,Lq be a p-local compact group. We define the Eulercharacteristic of the linking system L to be

�pLq “ limkÑ8

�pCk

q

where Ck

is the subcategory of L as defined in 4.4.

Example 4.6. Let S “ D28“ xx, t

n

| x2“ t2

n

n

“ 1, t2n`1 “ t

n

, xtn

x “ t´1n

y, and fixV “ xx, t1y. The objects of rL‚

s are S and V . The morphism sets are as follows

rL‚s C

k

AutpSq “ S AutpSq “ Sk

“ Dk`12

AutpV q “ ⌃4 AutpSq “ ⌃4

MorpV, Sq “ tx ˝ ↵ | x P S,↵ P t1,�,�2uu MorpV, Sq “ tx ˝ ↵ | x P S

k

,↵ P t1,�,�2uu

From the table we have the zeta matrix

Z “

ˆ24 2k`1

¨ 30 2k`1

˙

and consequently the weighting

k‚“

ˆkV

kS

˙“

ˆ´

18

12k`1

˙.

Using definition 2.8, we have �pCk

q “

12k`1 ´

18 .And now by definition 4.5 we get

�pLq “ limkÑ8

1

2k`1´

1

8“ ´

1

8

In order for definition 4.5 to be of much use, the limit must exist. Moreover, forit to be a reasonable definition, the limit should be independent of the choice ofunstable Adams operation. We will prove these facts in the next theorem.

Theorem 4.7. Let pS,F ,Lq be a p-local compact group and an unstable Adams

operation. If tCk

u

k•0 are fixed points of L under , then the limit limkÑ8 �pC

k

q

exists. Moreover, it is independent of the choice of Adams operation.

Before we prove this, we point out the following fact, which follows from the⇣-power action of on T :

Proposition 4.8. Let T “ pZ{p8Zq

r be the maximal torus of S. For any unstableAdams operation , there is a strictly increasing sequence of natural numbersn1, n2, . . . such that the group of fixed points of p

k

in T is pZ{pnkZq

r.

Next note that for any P and Q in ObL, Q has a natural left action onMorLpP,Qq defined by q ¨f “ �

Q

pqq ˝f . This parallels the left action of P – �P

pP q

on MorLpP,Qq discussed in (A) of the definition.


Lemma 4.9. Let P,Q be objects of a centric linking system L associated to a fusion

system F over S. Then the set of orbits QzMorLpP,Qq under the action is finite.

Proof. Since L is a linking system over F , we have a surjection

⇢ : MorLpP,Qq Ñ MorF pP,Qq,

and a surjection⇡ : MorF pP,Qq Ñ InnpQqzHomF pP,Qq

by projection. But InnpQqzHomF pP,Qq is finite [2, 2.5], so we can choose repre-sentatives 1, . . . , n

P MorLpP,Qq.Now for any ' P MorLpP,Qq, it must have the same image under ⇡ ˝ ⇢ as some

j

. Hence ⇢p'q is in the same orbit as ⇢p j

q, so there is some ch

P InnpQq suchthat ⇢p'q “ c

h

˝ ⇢p j

q “ ⇢p�Q

phq ˝ j

q by (B).But by (A), ⇢ induces a bijection between MorLpP,Qq{�

P

pZpP qq and HomF pP q.Hence ' and �

Q

phq ˝ j

are in the same orbit, so there is some �P

pzq P �P

pZpGqq

such that ' “ �Q

phq ˝ j

˝ �P

pzq.Finally we use (C) to replace

j

˝ �P

pzq with �Q

p⇢p j

qpzqq ˝ j

. Thus ' “

�Q

ph⇢p j

qpzqq ˝ j

“ h⇢p j

qpzq ¨ j

.Hence each orbit of QzMorLpP,Qq must contain some

j

, so there are finitelymany orbits. ⇤

Finally, we prove an important technical lemma on which the proof will rely.

Lemma 4.10. Let L, Ck

be as in Theorem 4.7. For each pair of objects P and Qof L, there is a constant cpP,Qq such that for large enough k,

|MorCkpP,Qq| “ cpP,Qqpnk rkQ.

Proof. We can choose representatives of the finitely many orbits of QzMorLpP,Qq.Each is in

MorLpP,Qq “

8§

i“1

MorCipP,Qq,

so it must be in some MorCipP,Qq. Choosing k larger than the maximum of the i,MorCkpP,Qq includes all the objects.

Now considering the fixed subgroup Qk

“ Q X Sk

§ Q, we see that by functori-ality of our Adams operation, the action of Q

k

must fix the subset MorCkpP,Qq Ñ

MorLpP,Qq, so there is an induced left action of Qk

on MorCkpP,Qq. Since repre-sentatives of all of the orbits are included, we see that there is a bijection

QzMorLpP,Qq – Qk

zMorCkpP,Qq

Similarly, we can choose representatives of the finite group ⇧ “ S{T , and thosecontained in Q are representatives of the quotient by Q’s maximal torus, ⇧pQq “

Q{T pQq § ⇧. Hence for k large enough that all these representatives are includedin S

k

, we have that the quotient

Qk

{pT pQq X Sk

q “ ⇧pQq

is fixed.Finally, since T pQq – pZ{p8Zq

rkQ, it is clear that T pQq X Sk

– pZ{pnkZq

rkQ,since T pQq § T . Hence |T pQq X S

k

| is pnk rkQ.Then setting cpP,Qq “ |⇧pQq||QzMorLpP,Qq|, we have |MorCkpP,Qq| “ cpP,Qqpnk rkQ

for large enough k. ⇤


Using this lemma, the proof of Theorem 4.7 is not too di�cult:

Proof. Consider the incidence matrices Mk

of the Ck

, indexed by the objects of Ck

.By Corollary 4.10, we know the elements are Mk

PQ

“ cpP,Qqpnk rkQ.Now consider the matrixM whose elements corresponding to P and Q are simply

the cpP,Qq, and consider a weighting on this matrix, i.e. a solution w to Mw “ 1.For each large enough k, we can define wk by wk

Q

“ wQ

pńk rkQ.

Then each element of Mkwk corresponding to P isÿ

QPOb Ck

|MorCkpP,Qq|wk

Q

“

ÿcpP,Qqpnk rkQw

Q

pńk rkQ

“

ÿcpP,Qqw

Q

“

ÿM

PQ

wQ

“ 1

so wk is a weighting for Ck

. Now

limkÑ8

�pCk

q “ limkÑ8

ÿ

QPOb Ck

wk

Q

“ limkÑ8

ÿ

QPOb Ck

wQ

pńk rkQ

“

ÿ

QPObrL‚ks

limkÑ8

wQ

pńk rkQ

“

ÿ

rkQ“0

wQ

,

and this finite sum clearly exists.To see that this is independent of our choice of Adams operation, we simply see

that the limit is dependent only the wQ

, which are determined from the matrix M ,i.e. on the cpP,Qq “ |⇧pQq||QzMorLpP,Qq|. It is clear that these are properties ofC only, and not of our unstable Adams operation. ⇤

Thus we may define the Euler characteristic �pLq, as

�pLq “ �prL‚sq “ lim

kÑ8�pC

k

q.

4.3. Fusion and Transporter Categories. Note that the only properties of Lthat we used were finiteness conditions: the finiteness of the objects of rL‚

s, andthe finiteness of the orbit space by action of subgroups. Both of these were simplyinherited from F . In particular, Proposition 4.2 tells us that F enjoys the firstproperty, implying that L and more generally any transporter system also satisfiesit. Meanwhile, [2, 2.5] tells us that F has the second property, and we proved inLemma 4.9 that this also descends to L, or indeed any transporter system. Thusthis theorem actually extends to defining an Euler characteristic on any transportersystem or saturated fusion system.

This result is not emphasized, as it is not clear that the Euler characteristic (orthe classifying space) of a fusion system bears any relevance to the p-local finitegroup.

5. Future Work

We have given a definition for the Euler characteristic of a p-local compact groupG “ pS,F ,Lq, and presented some evidence that this definition is a reasonable one– but further argument is needed to justify that this is the best or only definition.


Proving that the limit of the Euler characteristics of our finite approximationsexists is a step in the right direction; but we still could compare our definition toother possible definitions. Specifically, while we said that if �pLq should morallybe the same as �prL‚

sq, it remains to be shown that our approximation of L bysubcategories,/ of rL‚

s was the right choice. An alternative approximation wouldbe to instead directly approximate L or L‚, and take the limit of these Eulercharacteristics. We conjecture that there is a relationship between lim

kÑ8 �pCk

q tolim

kÑ8 �prL‚k

sq, where rL‚k

s is the approximation of L we get by directly taking thefixed points of L under an unstable Adams operation. In particular we would likethis to be an equality, but we hypothesize that the approximation we have providedis at least an upper bound for these limits.

L rL‚s C

k

rp´q‚s fixed points of

L Lk

rL‚k

s

rp´q‚sfixed points of

Conjecture 5.1. �prL‚k

sq § �pCk

q for all k

This conjecture would be useful in the sense that it would show that our defini-tion of Euler characteristic is in some way universal, in that it bounds any otherapproximation.

We believe that this conjecture is related to another open problem: the re-lationship between �pT

S

pGqq and �pTS

1pHqq when G § H. We conjecture that

�pTS

pGqq • �pTS

1pHqq.

Another body of work that remains to be done is to find direct formulas for theEuler characteristic directly in terms of L, or even in terms of the group if thep-local finite group comes from one, without requiring the limit. Such formulas, inthe spirit of [6], would allow for practical computation of the Euler characteristicof infinite p-local compact groups.

A. Related Work

Condition (III) for a fusion system to be saturated states:

(III) If P0 § P1 § P2 § . . . is an increasing sequence of subgroupswith union P , and f P HompP, Sq is any morphism such that eachf |

Pi P HomF pPi

, Sq, then f P HomF pP, Sq.

This makes too much use of the fact that homomorphisms live inside some largermorphism set HompP, Sq. Can we rephrase this without that fact?

(IIIa) If P0 § P1 § P2 § . . . is an increasing sequence of subgroupswith union P , and f

i

P HomF pPi

, Sq are morphisms such that


fi

|

Pj “ fj

whenever i ° j, then there is a unique f P HomF pP, Sq

such that for each i, f |

Pi “ fi

.

Note that this definition still implicitly uses the larger sets HompP, Sq, but onlyin defining restriction maps. This, we will see, is fine – restriction maps are part ofthe natural language of sheaves.

Claim A.1. Conditions (III) and (IIIa) are equivalent.

Proof. This will follow from the fact that (IIIa) is true if F is the full subcategoryof subgroups of S and homomorphisms between them. This we see by definingf : P Ñ S by fpxq “ f

i

pxq, which is well-defined since the fi

agree on intersections.Moreover, if x, y, P P , then they are in some P

i

, so fi

pxqfi

pyq “ fi

pxyq, and f is ahomomorphism. Finally, for any other g satisfying these restrictions and any x P P ,gpxq “ f

i

pxq “ fpxq, so f is unique.Now suppose claim (III) holds, and suppose we have an agreeing family f

i

as inthe statement of (IIIa). Then we have a unique f P HompP, Sq which restricts toeach f

i

. By (III), f P HomF pP, Sq, so (IIIa) holds.Conversely suppose (IIIa) holds and f P HompP, Sq satisfies the restriction con-

ditions. Then there is a unique g P HomF pP, Sq such that each g|

Pi “ f |

Pi . Byuniqueness in our fact, g “ f , and f P HomF pP, Sq, so (III) holds. ⇤

Thus we have rephrased (III) to be a unique gluing axiom. In this sense, it’srather like arguing that functions out of some open cover of a topological spacewhich agree on intersections glue together into functions on the whole space. Butwhat sort of topological space could look like subgroups? They’re not even closedunder union!

A.1. Grothendieck Topologies. Grothendieck topologies are categories that insome sense “look like” the category of open sets in a topology. They are usefulbecause they allow us to use geometric intuition to describe things we’d like tothink of geometrically but where there’s no actual topology – as long as there’s anotion of restriction, and of “covering”. We’ve already established that we haverestrictions between subgroups. How about covering?

Consider the motivation of chains. These are actually stronger than simplycovers (by generating set or by union). Specifically, in a filtration, every finitelygenerated subset is eventually entirely contained in some element of the filtration.

We’re now ready to move on to constructing a Grothendieck topology that cap-tures this notion of covering.

We will present the definitions of Grothendieck topologies interweaved with con-structing our particular one. So to start, we’ll say that a Grothendieck topology isconstructed over a fixed category. The category we’ll be using is the category S ofsubgroups of S, with exactly one morphism (the inclusion) from P to Q i↵ P § Q.This is the lattice of subgroups of S.

Definition A.2. Let C be a category, and c an object. A sieve on c is a subfunctorof hC

c

“ HomCp¨, cq. That is, it is a collection of morphisms to c which is stableunder precomposition in C.

Note that in S, sieves over P pick out collections of subgroups of P which aredownward stable. Of particular interest to us will be sieves on S which, like thefiltrations we discussed, contain elements above every finitely generated subgroup:


Definition A.3. A sieve C on P is a compactly cofinal covering if for anyfinitely generated Q, CpQq “ h

P

pQq.

(A note on terminology: we use this name because finitely generated subgroupsare compact elements in the lattice of subgroups, and the covers are cofinal at thecompact elements in the sense that they contain an element above each compactelement. Note that this definition, and the results of this section, generalize to anyalgebraic lattice.)

Now let’s move on to some properties that open covers have that we’d like togeneralize in our Grothendieck topologies. The first is that covers should inducecovers of open subsets, if we intersect each element with the subset. To formalizethis, we’ll need to make rigorous how this idea of intersection generalizes.

Definition A.4. Let C be a category, C a sieve on c P Ob C, and f : d Ñ c amorphism of C. Then the pullback sieve f˚C of C is the sieve on d consisting ofthose morphisms g : e Ñ d such that f ˝ g : e Ñ c is in C. If C is a poset, this isthose elements of C which lie below d.

Claim A.5. Pullbacks of compactly cofinal covers are compactly cofinal covers.

Proof. Let C be a compactly cofinal cover on P , and Qf

›Ñ P a subgroup. If Rg

›Ñ Qis finitely generated, then f ˝ g P CpRq so R P f˚CpRq. ⇤

Finally, we can define a Grothendieck topology in full.

Definition A.6. Let C be a category. A Grothendieck topology on C is a col-lection of distinguished “covering sieves” on each object of C, meeting the followingaxioms:

(T1): If C is a covering sieve on X and i : Y Ñ X is a morphism, then i˚Cis a covering sieve on Y .

(T2): Let C is a covering sieve on X, and D be a (not a priori covering) sieveon the same object X. If for each f : Y Ñ X in C, the pullback f˚D is acovering sieve on Y , then D is a covering sieve on X.

(T3): hX

is a covering sieve on X.

The Grothendieck topology on S we’ll define will be, unsurprisingly, the collec-tion of compactly cofinal covers. Now we’ve already shown that (T1) holds! (T3)is even easier: h

X

pP q “ hX

pP q regardless of whether P is finitely generated.

Claim A.7 (T2). Compactly cofinal covers are transitive.

Proof. Let C be a compactly cofinal cover of P , and D any sieve. Suppose all thepullbacks of D by morphisms in C are compactly cofinal covers.

For any finitely generated Qi

›Ñ P , i P CpQq, so consider the pullback sieve i˚D.

Qid

›Ñ Q is in i˚QpQq because i˚Q is compactly cofinal cover and Q is finite. Thussome h with h ˝ id “ i is in DpQq, i.e. i P DpQq. ⇤

Hence we have successfully defined a Grothendieck topology.

A.2. Sheaves on a Site. A site is simply a category endowed with a Grothendiecktopology. Thus we have a site corresponding to S with the compactly cofinaltopology.


Definition A.8. Let S be a site. A sheaf on S is a contravariant functor F fromS to the category of sets, such that for any covering sieve C on an object X, thenatural map HomphS

X

, F q Ñ HompC,F q is a bijection.

Now, this definition requires some serious unrolling. First, think of F as afamily of functions on each subgroup of S, along with restriction maps betweenthem. Morphisms to F are actually natural transformations, and all the sourceshSX

pP q or CpP q have one or zero objects. Thus they’re ways of assigning to each ofthe picked-out subgroups a “function”, such that all the restriction maps commute.Hence we can think of natural transformations hS

X

Ñ F as elements of F pXq (sinceall other elements are determined as the restriction of this), and general naturaltransformations C Ñ F as collections of elements of F pcq for c picked out by Cwhich agree on intersections.

Next, what is the natural map HomphSX

, F q Ñ HompC,F q? Recall that Cwas a subfunctor of hS

X

, and thus we have an inclusion natural transformationCpP q ãÑ hS

X

pP q. Thus we can consider a natural transformation from hSX

to F toactually be a natural transformation from C to F , by composition. In the previousinterpretation, this is interpreting elements of F pXq as elements of each F pcq, bytaking the restriction maps.

Thus, what a sheaf is requiring is that each collection of functions on any com-pactly cofinal cover yields a unique gluing on their union. Now, this looks an awfullot like our earlier conditions.

So let’s make a new condition. Here we’ll use that S is a subcategory of F ,containing all the objects, and only the inclusion morphisms.

(IIIb) Let R § S. Then the functor hFR

: F Ñ Set restricts to asheaf on S with the compactly cofinal topology.

As always, we’d like this to be equivalent to our other conditions. Fortunately,it is.

Proof. Assume (IIIb) and let P0 § P1 § . . . be a filtration of P . Then the sieve C onP consisting of all inclusions factoring through the P

i

is a compactly cofinal cover:for each finite subgroup F “ tx1, . . . , xn

u we can choose Pi

containing each xi

, andthe maximum of these contains F , so F ãÑ P

i

ãÑ P is in CpP q. Now collectionsof functions on the P

i

which agree on intersections form a natural transformationfrom this cover to hF

S

|S . By (IIIb) they induce a natural transformation from hX

to hFS

|S , i.e. an element of hFS

pXq “ HomF pX,Sq.Conversely, assume (IIIc) and let C be any compactly cofinal cover. Order

the countably many elements of P as p0, p1, . . . (since p-toral groups are count-able). Choose P

i

“ xx1, . . . , xi

y and note that CpPi

q is nonempty (since C is acompactly cofinal cover and P

i

is finitely generated). Then any natural transfor-mations C Ñ hF

S

|S induces maps fi

P hFS

pPi

q “ HomF pPi

, Sq which agree onintersections. By (IIIb) these glue together to a map f P HomF pP, Sq “ hF

S

pP q

which restricts appropriately on the Pi

. Thus our natural transformation induces anatural transformation hS

P

Ñ hSR

|S . Since this holds for any P , hFS

|S is a sheaf. ⇤

So perhaps a better geometric view of a saturated fusion system pS,Fq is as thepair pS, hF

S

q of a site and a structure sheaf on it. This itself is a very geometricdefinition: it parallels the definition of a manifold or a scheme, and echoes the ideathat we would like to think of a fusion system as a geometric object. Moreover,


from this pair we can reconstruct F by placing all the functions in this sheaf intoS as morphisms to their images, and extending by composition.

The definition of a saturated fusion system over S then becomes simple: it isa sheaf of injective group homomorphisms into S, satisfying a relative finitenesscondition on its global sections and an extension condition.

Note that this definition allows us to reinterpret some of the existing theoryaround fusion systems. For example, Alperin’s Fusion Theorem tells us that anysection in this sheaf extends to a section on a centric radical subgroup. It is this factthat allows us, when constructing the linking system, to discard subgroups belowthese, and still be confident that the resulting topology is fine enough to generatethe full sheaf.

A.3. Final Thoughts. It would be interesting to see if this reinterpretation ofsaturated fusion systems as geometric objects allows for any extension of theirexisting theory, or for interesting new proofs of their properties. This seems like anarea ripe for future development.

It should be pointed out that we could view the sheaves in F as sheaves overa topology on F , instead of on S. Namely, we can define the compactlyf cofinaltopology in exactly the same way as we did on S – motivating why we madesometimes opaque definitions such as “CpRq “ h

S

pRq” instead of “the inclusionof R is in C.” It is unclear to the authors which site is a more natural settingfor this sheaf. On F the construction would have possibly been cleaner, requiringonly that representable functors be sheaves (without restrictions), and F embedsdirectly into its own topos of sheaves. On the other hand, S is a more intuitivetopology, and expresses our notion that the fusion system is “on S.”

It would also be interesting to see if this point of view extends to linking systems,or general transporter theorems. Another direction would be to develop the theoryof morphisms between saturated fusion systems in this view – does the intuitiveidea of a morphism of sites along with a pullback or pushforward map of sheavescorrespond to the preexisting notion of morphisms of saturated fusion systems?

Finally, since we have a translation of groups into generally geometric objects, itwould be good to know if there is any relationship between the geometry of theseobjects and of the classifying space, either of the group or the linking or fusionsystem.

References

[1] A. K. Bousfield and D. M. Kan. Localization and completion in homotopy theory. Bulletin of

the American Mathematical Society, 77(6):1006–1010, 11 1971.[2] Carles Broto, Ran Levi, and Bob Oliver. Discrete models for the p-local homotopy theory of

compact lie groups and p-compact groups. Geom. Topol, 11:315–427, 2007.[3] Carles Broto, Ran Levi, and Bob Oliver. An algebraic model for finite loop spaces. arXiv

preprint arXiv:1212.2033, 2012.[4] Charles Broto, Ran Levi, and Bob Oliver. The homotopy theory of fusion systems. Journal of

the American Mathematical Society, 16(4):779–856, 2003.[5] Alex Gonzalez. Unstable adams operations acting on p–local compact groups and fixed points.

Algebr. Geom. Topol, 12(2):867–908, 2012.[6] Martin Wedel Jacobsen and Jesper M Moller. Euler characteristics of p-subgroup categories.

arXiv preprint arXiv:1007.1890, 2010.[7] Fabien Junod, Ran Levi, and Assaf Libman. Unstable adams operations on p–local compact

groups. Alg. Geom. Topol, 12:49–74, 2012.[8] Tom Leinster. The euler characteristic of a category. Doc. Math, 13(21-49):119, 2008.


[9] Gian-Carlo Rota. On the foundations of combinatorial theory i. theory of mobius functions.Probability theory and related fields, 2(4):340–368, 1964.

(Andrew George Batallas) Vanderbilt University, Nashville, TN 37235

E-mail address: [email protected]

(Alex Gonzalez) Kansas State University, Manhattan, KS 66506

E-mail address, Corresponding author: [email protected]

(Mitchell Messmore) University of Redlands, Redlands, CA 92373

E-mail address: mitchell [email protected]

(Andrew Smith) Carnegie Mellon University, Pittsburgh, PA 15213

E-mail address: [email protected]

EULER CHARACTERISTICS OF p-LOCAL COMPACT GROUPS€¦ · 2. The Euler Characteristic of Finite Categories 3 3. p-local compact groups 5 3.1. Discrete p-toral groups and fusion systems

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