Homology decompositions for classifying spaces of finite groups - - PDF document

Homology decompositions for classifying spaces of finite groups

Silvia Onofrei Department of Mathematics, University of California at Riverside

Thursday, October 20, 2005

• 1. Outline of the talk and motivation

Let G be a finite group and p a prime dividing its order. A homology decomposition of a classifying space BG is a process of glueing together classifying spaces of suitable chosen subgroups

• f G in order to obtain a space X and a map X → BG which induces an isomorphism in mod-p

homology. The glueing information is encoded in the form of a homotopy colimit. The mod-p homology of BG can then be determined by an exact sequence of the homology groups of the subgroups if certain higher derived functors associated to the homotopy colimit are zero. Homology decompositions provide a systematic way for approaching the cohomology of simple groups. The finite simple groups can be divided into three classes: the alternating groups (for n ≥ 5), the groups of Lie type and the 26 sporadic groups. Results obtained in the past two decades show that the mod p (especially for p = 2) cohomology of the sporadic groups exibit interesting features and finding the cohomology of these groups remains an area of active research. In an attempt to generalize the theory of buildings for Lie groups, several p-local geometries (geometric structures which have the property that the stabilizers of the objects are normalizers

• f p-subgroups of the group G) have been constructed for the sporadic finite simple groups but

there is no uniform treatment of the p-local geometries of these groups. However, many sporadic groups, have a p-local geometry whose barycentric subdivision corresponds to a certain family of p-subgroups called the Bouc collection (this contains all the p-subgroups such that NG(P)/P does not have non-trivial normal p-subgroups). A very recent work of Benson and Smith [BS04] focuses on the p-local approach to the structure

• f the mod-p cohomology of the sporadic groups. For p = 2, they exhibit for each sporadic group

an ample collection of 2-subgroups related to a 2-local geometry. By a case by case analysis, they

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describe homotopy equivalences between most of these collections and one of the standard collections, such as the Quillen or the Bouc collection. Our work (joint with J. Maginnis) began with an analysis of the homotopy relation between a 2-local geometry and the aforementioned Bouc collection for the third sporadic group of Conway. We were able to define a new collection which is in the same homotopy class as the 2-local geometry. Motivated by this result we generalized the definition of this collection. The novelty of our approach resides in the fact that we impose conditions on the groups which are studied, instead of imposing conditions on their collections of subgroups. In this talk I shall present these new collections of p-subgroups which can be used to obtain homology decompositions for finite groups. For the 26 sporadic simple groups these collections are related to p-local geometries. I shall start with few definitions, followed by a brief outline of the development of the area from historic point of view; then continue with the simple example of the group GL(3, 2). Next, I shall adumbrate a general description for the construction of the homotopy decomposition adapted for the case of the subgroup decomposition. Finally the statement of the main theorem and an outline

• f its proof will be given.
• 2. Some terminology

A collection C is a family of subgroups of G which is closed under conjugation and ordered by inclusion; thus a collection is a G-poset, under the conjugation action of the group. To this G-poset

• ne can associate a G-simplicial complex (and thus a topological space) |C| whose vertices are the

elements of C and whose simplices are the nonempty finite chains in C. This is called the subgroup complex associated to C. In finite group theory, the subgroup complexes play an important role, they are regarded as geometries and generalizations of buildings. If C is viewed as a small category, then |C| is the classifying space of C; the functor C − → |C| assigns topological concepts to a posets. Probably the best known group complexes are the buildings. If we let G to be a Lie group in natural characteristic p, then the building of G is the simplicial complex of the poset of parabolic subgroups (the overgroups of the Borel subgroup). The theory of buildings was initiated by Tits [T74] in order to give a systematic approach for the geometric interpretation of the semisimple Lie groups and in particular those of exceptional type. The understanding of groups is simplified through the use of geometries on which these groups act and the theory of buildings provides a tool in approaching these geometries directly. The topology of buildings is relevant to the representation theory of the underlying Lie group: the top dimensional homology module is the Steinberg module, the only irreducible projective module of G.

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The following collections are standard in the literature: Ap(G) = {E | E nontrivial elementary abelian p-subgroup of G}; Sp(G) = {P | P nontrivial p-subgroup of G}; Bp(G) = {R | R nontrivial p-radical subgroup of G}. The simplicial complex |Ap(G)| is known as the Quillen complex, |Sp(G)| is known as the Brown complex and |Bp(G)| is called the Bouc complex. The three complexes are G-homotopy equivalent [ThWb91].

• 3. Historical context

Simplicial complexes from posets of subgroups of a group Brown [Br75, Br76] studied the influence of the elements of finite p-power order on group cohomology1, in groups which were not necessarily finite.2 Restricted to the case of a finite group G, his work focused on the poset Sp(G) of all nontrivial p-subgroups of G ordered by inclusion and its associated simplicial complex |Sp(G)|, obtained from the inclusion chains of p-subgroups. One

• f the most interesting results he obtained refered to the Euler characteristic of this complex:

χ(|Sp(G)|) ≡ 1 (mod|G|p) where |G|p is the highest power of p which divides the order of G, thus the order of a Sylow p- subgroup if G is finite. This is also known as the homological Sylow theorem, due to the resemblance to the Sylow theorem from group theory which affirms that the number of Sylow p-subgroups of a finite group G is congruent to 1 mod p. Quillen [Qu78] studied in his seminal paper the homotopy properties of Sp(G). He developed the fundaments for the study of the homotopy properties of posets of subgroups. In fact, Quillen introduced the poset of non-trivial elementary abelian p-subgroups, which is usually denoted Ap(G) and it is known today as the Quillen poset. He proved that the inclusion of posets Ap(G) ⊆ Sp(G) induces a homotopy equivalence. Another common collection was introduced by Bouc [Bou84]; this collection consists of non- trivial radical p-subgroups (also known as p-stubborn subgroups). These are p-subgroups R with the property that NG(R)/R does not have nontrivial normal p-subgroups. Bouc proved that this new collection has the same homotopy type as the previously defined collections of Brown and Quillen. For a finite Chevalley group in characteristic p, the Bouc collection is the collection of unipotent

1In fact, he was interested in the cohomology of the group and the relation to the p-part of the Euler characteristic.

2Brown considered infinite groups, in particular he was interested in discrete groups of virtually finite cohomolog-

ical dimension, but we will focus on finite groups only.

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radicals of parabolic subgroups, and it corresponds to the barycentric subdivision of the associated Tits building. It was later observed by Th´ evenaz and Webb [ThWb91] that the homotopy equivalences between these three collections are in fact G-homotopy equivalences and most of their properties can be rewritten in G-equivariant form. A comprehensive study of the homotopy equivalences between nine standard collections is pre- sented in Grodal and Smith [GrS04]. Homology decompositions The first result to be mentioned here is Webb’s alternating-sum formula for group cohomology [Wb87]. One version of this theorem states that if G is a finite group acting on a simplicial complex ∆ such that the fixed point sets ∆P of all subgroups P of order p are contractible3, then: H∗(G; Fp) =

• σ∈∆/G

(−1)dimσH∗(Gσ; Fp) where Gσ is the isotropy group of the simplex σ. Note that the above direct sum is indexed on the simplices of the orbit complex. This formula is valid for the three complexes mentioned above and for the Tits building of a finite Chevalley group in defining characteristic. This formula gives one of the first known homology decompositions for B at p. Homology decompositions of classifying spaces of compact Lie groups were exploited by Jack-

• wski, McClure and Oliver [JM92, JMO92] and others as a powerful tool. For example Jackowski

and McClure [JM92] approximate BG by classifying spaces of centralizers of non-trivial elemen- tary abelian p-subgroups of G. Jackowski, McClure and Oliver [JM92] approximate BG, for G a compact Lie group, by classifying spaces of p-radical subgroups of G. Later Dwyer [Dw97, Dw98, DwH01] gave a systematic approach for homology decompo- sitions of classifying spaces of finite groups. Dwyer described three different types of homology decompositions known as subgroup, centralizer and normalizer decompositions. Every collection of subgroups of a finite group G gives rise to three different decompositons, one of each type. Dwyer introduced the concepts of ampleness and sharpness for a collection C of p-subgroups of a finite group G. A decomposition is sharp if certain higher derived functors vanish, thus yielding a partic- ularly simple formula for the cohomology of BG in terms of the cohomology of the spaces involved in the decomposition. Sharpness depends on the collection and the type of decomposition chosen. For example, the collection of non-trivial elementary abelian p-subgroups of G is centralizer and normalizer sharp but not subgroup sharp.

3A later version of this theorem requires acyclicity of the fixed point sets.

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• 4. An example: the group GL(3, F2)

Let G = GL(3, F2), the group of 3 × 3 matrices with entries from the field with two elements. This is a simple group of order 168, whose Sylow 2-subgroup is the dihedral group D8. Tits building: the extrinsic approach. Let V = F3

2 be the 3-dimensional vector space over

• F2. The group G acts naturally by right multiplication on the row vectors. The finite geometry for G

is the projective plane of the natural module V . The projective points are 1-dimensional subspaces

• f V and the projective lines are the 2-dimensional subspaces of V . There are 7 projective points

and 7 projective lines. Each line contains three points and each point is on three lines; therefore there are 21 flags (inclusion chains of the type 1-subspace ⊂ 2-subspace). As a simplicial complex: there are 7 + 7 vertices and 21 edges. The group G acts transitively on the maximal faces of this complex. Stabilizers Gp = SI

4 = 22 I.S3 = NG(Z2 × Z2)

Gl = SII

4 = 22 II.S3 = NG(Z2 × Z2)

Gpl = D8 = 21+2 = NG(Z2)

Picture: Fano plane

Tits building: the intrinsic approach. We consider the collection of the parabolic subgroups

• f G, the overgroups of the Borel subgroup B = D8. In fact, each of the parabolic subgroups is

G-conjugate to one of the simplex stabilizers from above. Note that there are two conjugacy classes

• f S4’s. This is expressed in the following diagram:

S4

• D8
• S4

The Webb alternating sum formula, written as an exact sequence for cohomology reads: 0 → H∗(G, F2) → H∗(S4, F2) ⊕ H∗(S4, F2) → H∗(D8, F2) → 0 Remark 4.1. The cohomology of D8 with F2 coefficients is H∗(D8, F2) ≃ F2[x, y, ω]/xy = 0, where x and y are 1-

dimensional generators and ω is a 2-dimensional generator. The cohomology of GL(3, 2) is given by H∗(G, F2) = F2[α, β, γ]/βγ. Since the cohomology of a group with Fp coefficients injects into the cohomology of the Sylow p-subgroup via the restriction map H∗(G, F2) res − → H∗(D8, F2) the generators of H∗(G, F2) map as follows: α − → ω + x2 + y2 β − → xω γ − → yω

6 For the cohomology of S4 consider the sequence D8 − → S4 − → GL(3, 2) Again, the cohomology of S4 injects in the cohomology of D8 and we can describe the latter as H∗(S4, F2) = F2[m, n, q]/mq = 0 with the following correspondence under the restriction map: m − → x n − → ω + y2 q − → yω Recall now that G = GL(3, 2) contains two conjugacy classes of S4’s, say SI

4 and SII 4 . If the cohomology of SI 4 is given as above,

then H∗(SII

4 , F2) is obtain by interchanging x with y.

The Quillen complex. There are three conjugacy classes of elementary abelian 2-subgroups in G which can be written as 2, 22

I and 22

• II. Then |A2(G)| is a 2-dimensional complex whose maximal

faces are 2 < 22

I and 2 < 22 II and NG(2 < 22 I) ≃ NG(2 < 22 II) ≃ D8.

NG(2) ≃ D8 NG(22

I) ≃ SI 4

NG(22

II) ≃ SII 4

NG(2 < 22

I) = NG(2) ∩ NG(22 I) ≃ D8

NG(2 < 22

II) = NG(2) ∩ NG(22 II) ≃ D8

Picture: The orbit complex of the Quillen complex A2(G)/G.

There is an exact sequence in cohomology: 0 → H∗(G, F2) → H∗(D8, F2) ⊕ H∗(S4, F2) ⊕ H∗(S4, F2) → H∗(D8, F2) ⊕ H∗(D8, F2) → 0 Note that the above exact sequence gives the cohomology of the group in term of the cohomology

• f the normalizers of the subgroups in the orbit complex of the Quillen complex. For this reason,

this decomposition is called the normalizer decomposition. Since the sequence is exact we say that A2(G) is normalizer sharp. In fact, the Quillen complex turns out to be normalizer sharp for any finite group G. Also note that, in the alternating sum two terms for D8 cancel and we obtain the first exact sequence. If we consider the centralizers of the simplices in the Quillen complex, which are CG(2) ≃ D8, CG(22) ≃ 22 and CG(2 < 22) = CG(2) ≃ D8, we obtain the exact sequence in cohomology: 0 → H∗(G, F2) → H∗(D8, F2)D8 ⊕ H∗(22, F2)S4 ⊕ H∗(22, F2)S4 → H∗(22, F2)D8 ⊕ H∗(22, F2)D8 → 0 Since a group acts trivially on its own cohomology we obtain: 0 → H∗(G, F2) → H∗(D8, F2) ⊕ H∗(22, F2)S3 ⊕ H∗(22, F2)S3 → H∗(22, F2)2 ⊕ H∗(22, F2)2 → 0 This decomposition is called the centralizer decomposition. Since the sequence is exact we say that A2(G) is centralizer sharp. The Quillen complex is centralizer sharp for any finite group G.

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Finally, let us write a similar sequence using the subgroups of the Quillen complex. We get: 0 → H∗(G, F2) → H∗(Z2, F2)D8 ⊕ H∗(Z2 × Z2, F2)S4 ⊕ H∗(Z2 × Z2, F2)S4 → → H∗(Z2, F2)D8 ⊕ H∗(Z2, F2)D8 → 0 which can be rewritten as: 0 → H∗(G, F2) → H∗(Z2, F2) ⊕ H∗(Z2 × Z2, F2)S3 ⊕ H∗(Z2 × Z2, F2)S3 → → H∗(Z2, F2) ⊕ H∗(Z2, F2) → 0 Next consider the following map: ϕ : H1(Z2, F2) ⊕ H1(Z2 × Z2, F2)S3 ⊕ H1(Z2 × Z2, F2)S3 → H1(Z2, F2) ⊕ H1(Z2, F2) Since the H1(Z2, F2) ≃ Z2 and H∗(Z2 × Z2, F2)F2[a, b] is a polynomial algebra in two generators, both 1-dimensional. But S3 permutes the three 1-dimensional elements a, b, a + b and thus H1(Z2 × Z2, F2)S3 = 0. Therefore we can rewrite: ϕ : Z2 → Z2 ⊕ Z2 which shows that this map cannot be surjective and thus the sequence is not exact. Therefore, the subgroup decomposition for the Quillen complex is not subgroup sharp. This is part of a more general phenomenon, the Quillen complex is not subgroups sharp. Remark 4.2. Note that the cohomology of GL(3, 2) is detected on elementary abelian subgroups, but the Quillen complex is not subgroup sharp in this case. The cohomology of G is detected on elementary abelian subgroups if for every nonzero x ∈ H∗BG there is an elementary abelian p- subgroup V of G such that the restriction of x to H∗BV is nonzero.

• 5. Homology decompositions

In what follows, I shall present a general construction, due to Dwyer, adapted to the case of subgroup decomposition for a collection of subgroups. Given a collection of subgroups, one tries to define a suitable category whose objects are transitive G-sets and such that the isotropy groups are either the subgroups themselves, the centralizers or the normalizers of these subgroups. These three different constructions give rise to the subgroup, centralizer or the normalizer decomposition. The general treatment is similar in all three cases.

• Notation. Let G be a finite group, p a prime divisor of its order and Fp the finite field with p
• elements. Let C be a collection of subgroups of G. Let BG be the classifying space of G and EG its

universal cover4. Let G-Set be the category of transitive G-sets, which we shall regard embedded

4The classifying space of G is a connected CW-complex with fundamental group G and higher homotopy groups

zero, thus an Eilenberg-MacLane space K(G, 1). The group G acts freely on its universal cover EG which is a contractible space. The homology and cohomology of G can be defined as the homology and cohomology of BG.

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in G-Sp, the category of G-spaces. For the purpose of this talk, a G-space is a CW-complex with a compatible cellular G-action5. For a G-space X let EG ×G X denote the corresponding Borel construction6. Homotopy colimit. Let (Xi, fij) be a diagram of spaces. Then the colimit of this diagram is the space defined by ∐Xi/{x ∼ fij(x)}, the qutient of the disjoint union of spaces Xi associated to the vertices of the diagram of spaces Γ, under identification x ∼ fij(x) where fij : Xi → Xj, for each edge in Γ.

Picture: Pushout square, the colimit and the homotopy colimit.

In order to obtain the homotopy colimit, consider the disjoint union of spacex Xi, regarded as the product of Xi with a 0-simplex. For each nonidentity morphism fij we take a copy of Xi × ∆1 (regard ∆1 = [0, 1]). The gluing: identify one end Xi × {0} with Xj × ∆0 and use the identity map on Xi to identify the other end Xi × {1} with Xi × ∆0. For a pair of composable arrows Xi → Xj → Xk we take a copy of Xi × ∆2 and make similar identifications. Iterate this process. The advantage of using homotopy colimits over ordinary colimits, is that the latter are homotopy

• invariants. See example below:

∗ − → X ← − ∗ CX − → X ← − CX For the first diagram the colimit is ∗ while the homotopy colimit is SX. For the second line, both the limit and the colimit are SX. The C-orbit category OC is the category whose objects are the G-sets G/H, H ∈ G and whose morphisms are G-maps.

5Note that any compact manifold with a G-action can be given the structure of a finite CW-complex. 6The Borel construction is the orbit space EG ×G X = (X × EG)/G where G acts diagonally on EG × X.

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Definition 5.1. A mod p homology decomposition of BG consists of a functor F : OC → G-Set with the property that the map hocolimOC(EG ×G F) → BG is a mod p homology isomorphism. Denote |EOC| = hocolimOC F, this is a space and it can be proved that it is isomorphic to the nerve of the category EOC, described below. Let EOC be the category whose objects are pairs (G/H, xH) where H ∈ C and xH ∈ G/H. A morphism (G/H, xH) → (G/K, yK) is a G-map f : G/H → G/K with f(xH) = yK. The G-action is given by g · (G/H, xH) = (G/H, gxH) and thus the isotropy groups are of the form xH = xHx−1 with H ∈ C. Let F : EOC → Sp be given by F = EG ×G

• F. For every G/H ∈ Obj(OC), the values are given

by F(G/H) = EG ×G G/H ≃ BH. We use the fact that the homotopy colimit commutes with the Borel construction: hocolimOC F ≃ hocolimOC(EG ×G F) ≃ EG ×G hocolimOC( F) ≃ EG ×G |EOC| The collection C is ample if and only if the map EG×G |EOC| → BG induces an isomorphism in mod p homology. The decomposition associated to this map is called the subgroup decomposition. A homotopy colimit gives rise to a Bousfield-Kan homology spectral sequence with E2

ij = colim OC iHj(F, Fp) ⇒ Hi+j(hocolimOCF; Fp) ≃ Hi+j(BG; Fp).

Definition 5.2. The homology decomposition from 5.1 is subgroup sharp if its Bousfield-Kan homology spectral sequence collapses at the E2-page onto the vertical axis. This can be rephrased as follows: colim

OC iHj(EG ×G

F; Fp) = 0 for i > 0 colim 0H∗(EG ×G F; Fp) ≃ H∗(BG; Fp). From the E2-page onward, the Leray spectral sequence of the map: ϕ : EG ×G |EOC| = hocolimOCF → hocolimOC(∗) ≃ |OC| is the Bousfield-Kan homology spectral sequence for hocolimOCF. The isotropy spectral sequence of a G-space X is the Leray spectral sequence of the map: ψ : EG ×G X → X/G We have the following: |EOC|/G =

• hocolimD

F

• /G = hocolimOC

F/G

• ≃ hocolimOC(∗) = |OC|.

Therefore the Bousfield-Kan homology spectral sequence can be identified, from the E2-page

• nward, with the isotropy spectral sequence associated to the action of G on X.

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The E1-page of the isotropy spectral sequence is E1

ij = Hj(G; Fp[Xi])

where Xi is the set of i-simplices of X. This can be regarded as the chain complex CG

n (X; H) for

the coefficient functor H(A) = Hj(G; A); for any Fp[G]-module A. The boundary maps are induced by the alternating sum of face maps in X. Thus the E2-page is isomorphic to the Bredon homology

• f X. The sharpness properties can be rephrased in terms of the Bredon homology and acyclicity
• f the G-space X.

Collections which are subgroup sharp are Sp(G) and Bp(G).

• 6. Distinguished collections of p-subgroups

Definitions and basic properties Let Γp(G) be the family of elements of order p in G which are conjugate to a central element, that is some element of central type in G. For a p-subgroup P of G define:

• P = x|x ∈ Ω1Z(P) ∩ Γp(G)

Further, for Cp(G) a collection of p-subgroups of G denote:

• Cp(G) = {P|P ∈ Cp(G) and

P = 1 } We call Cp(G) the distinguished Cp(G) collection. We shall refer to the subgroups in Cp(G) as distinguished subgroups. Remark 6.1. If G has one conjugacy class of elements of order p, it is obvious that Cp(G) = Cp(G). Clearly if P ≤ G is a p-subgroup then P ≤

• P. Also

Cp(G) ⊆ Cp(G). Before proceeding to the study of the distinguished collections of p-subgroups we formulate the following conditions: (M) Given any P ∈ Sp(G); the subgroup NG(P) is contained in a p-local subgroup which contains a Sylow p-subgroup of G. (Cl) The central elements of order p in G are closed under products of commuting elements. (Ch) The group G is of characteristic p-type. Remark 6.2. Note that if R is a p-subgroup of G and if NG(R) contains a Sylow p-subgroup of G, then R is distinguished. This is easy to see since, in this case R ⊳ S ∈ Sylp(NG(R)) ⊆ Sylp(G). Thus R ∩ Z(S) = 1.

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Remark 6.3 (The three conditions and the sporadic groups). The three conditions listed above were formulated with the properties of the 26 sporadic simple groups in mind. Let p = 2 and let S denote the set of the 26 sporadic groups. All but four of the elements of S satisfy at least one of the three assumptions.

Some homotopy equivalences.

Proposition 6.4. The inclusion Ap(G) → Sp(G) induces a G-homotopy equivalence on nerves.

• Proof. The proof is similar to that of Proposition 3.4, in which we replace “tilde” by “hat”.

For P ∈ Sp(G) and Q ∈ Ap(G)≤P the subposet Ap(G)≤P is NG(P)-contractible via the double inequality Q ≤ Q P ≥ P given by NG(P)-equivariant poset maps. The G-homotopy equivalence follows by an application of Theorem 2.6(ii).

• Definition 6.5. Let P ∈ Sp(G). Consider the subposet of distinguished p-subgroups defined

by N>P = {Q ∈ Sp(G) | P < Q ≤ NG(P)}. Lemmatta 6.6. Let P ∈ Sp(G) and Q ∈ Sp(G)>P . Then NQ(P) is a distinguished p-subgroup in G.

• Proof. Let P and Q be as in the hypothesis. Then P < NQ(P) ≤ Q. Also Z(Q) ≤ Z(NQ(P)).

Therefore NQ(P) ∈ N>P ⊆ Sp(G)>P .

• Lemma 6.7. For P ∈ Sp(G), the poset

Sp(G)>P is NG(P)-homotopy equivalent to the subposet N>P .

• Proof. Consider the poset map f :

Sp(G)>P → Sp(G)>P defined by f(Q) = NQ(P); note that f ≤ Id b

Sp(G)>P and that f(

Sp(G)>P ) = N>P . By Proposition 2.9 it follows that Sp(G)>P is homotopy equivalent to f( Sp(G)>P ). Since the map f is NG(P)-equivariant this is a NG(P)- homotopy equivalence.

• Proposition 6.8. Assume that G is a finite group with the property that one of the conditions

(M), (Cl) or (Ch) holds. If P ∈ Sp(G) is such that P < Op(NG(P)) then N>P is NG(P)-contractible.

• Proof. Let P ∈

Sp(G) \ Bp(G) and denote ONP = Op(NG(P)). Further, let Q ∈ N>P so P < Q ≤ NG(P). For ¯ S ∈ Sylp(NG(P)) with the property that Q ≤ ¯ S, let S denote a Sylow p-subgroup of G such that ¯ S = S ∩ NG(P). Note that since Z(S) ≤ CG(P) ≤ NG(P) it follows that Z(S) ≤ Z( ¯ S). (M). Let M a p-local subgroup of G with NG(P) ≤ M and such that |M|p = |G|p, that is the subgroup M has the same Sylow p-subgroup as G. Denote R = Op(M) and assume that S, the Sylow p-subgroup chosen in the previous paragraph also lies in M. Since R S the intersection R ∩ Z(S) is nontrivial. Consider the following string of inequalities: Q ≤ QNR(P)ONP ≥ NR(P)ONP

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Note Q normalizes NR(P) since R M ≥ Q and Q ≤ NG(P). Also Q normalizes ONP NG(P) ≥ Q. From the fact that 1 = Z(S) ∩ R ≤ Z(R) ≤ Z(NR(P)ONP ) it follows that NR(P)ONP is

• distinguished. Since 1 = Z(S) ∩ Z(R) ≤ Z(QNR(P)ONP ) it follows that this last p-subgroup is

also distinguished. Thus since all the inequalities correspond to NG(P)-equivariant poset maps N>P → N>P it follows that N>P is NG(P)-contractible. (Ch). Since G is of characteristic p-type CNG(P)(ONP ) ≤ ONP . Thus P < ONP ≤ ¯ S ≤ S so Z(S) ≤ CNG(P)(ONP ) ≤ ONP . Therefore Z(S) ≤ Z(ONP ). It follows that ONP ∈ Sp(G). Next, note that QONP is distinguished, which follows from the fact Z( ¯ S) ≤ Z(QONP ). Now consider the string of poset maps N>P → N>P : Q ≤ QONP ≥ ONP which proves the NG(P)-contractibility of N>P . (Cl). Consider the string of NG(P)-equivariant poset maps: Q ≤ QONP ≥ ONP In order to finish the proof we have to prove that ONP ∈ Sp(G). Note that P ⊳ONP so P ∩Z(ONP ) = 1 and using our assumption (Cl) we obtain that Z(ONP ) ∩ Γp = 1.

• Corollary 6.9. Assume that G satisfies one of the conditions(Cl), (Ch) or (M). Then

Sp(G) and Bp(G) are G-homotopy equivalent. Theorem 6.10. Let C be one of the collections Ap(G), Sp(G) or Bp(G). For simplicity we will

• mit the group G. Then there exist homotopy equivalences, summarized in the following table:
• Ap
• Sp
• Bp

|EOC|

• · · · · · · · · · · · · · · · · · ·
• (Cl, Ch, M)
• |C|
• .

. . . . . . . .

• |

|

(Cl, Ch) (Cl, Ch, M)

• |

|

(Cl, Ch)

|EAC|

• |

|

(Cl, Ch)

• |

|

(Cl, Ch)

· · · · · · · · · · · · · · · · · ·

• .

. . . . . . . . Notation 6.11. In the above table a solid line corresponds to a G-homotopy equivalence, a dashed line to S-homotopy equivalence and a dotted line to an ordinary homotopy equivalence. Here S denotes a Sylow p-subgroup of G. A label (c) means that the corresponding homotopy equivalence holds under hypothesis (c).

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• Proof. We first show how to obtain the horizontal lines. Let C′ =

Ap(G) and C = Sp(G). The solid horizontal line on the middle row follows from Proposition 4.8. The proof for the solid horizontal line between |EAC′| and |EAC| is similar to that in Theorem 3.1. Let H be a subgroup

• f G and denote X = C′

≤CG(H) and Y = C≤CG(H). We show that X≤P is CG(P)-contractible for any

P ∈ Y. For Q ∈ X≤P we obtain the contracting homotopy Q ≤ PQ ≥ P which proves the conical contractibility of X≤P , via poset maps which are CG(P)-equivariant. Now apply Theorem 2.6(i) to show that X and Y are G-homotopy equivalent and Remark 2.16 to obtain the G-homotopy equivalence between |EAC| and |EAC′|. Next consider C′ = Bp(G) and C = Sp(G). The middle solid horizontal line is given by Corollary 4.13. Let P ∈ C \ C′. From Lemma 4.11 and Proposition 4.12 we know that if G satisfies one of the hypotheses (Cl), (Ch) or (M), the subposet C>P is NG(P)-contractible. Next we apply Lemma 2.18(i) to obtain the G-homotopy equivalence between |EOC′| and |EOC|. To see that the two remaining dotted horizontal lines cannot be replaced by dashed lines, consider the example G = D8, discussed in the proof of Theorem 3.1. This group satisfies all three hypotheses (M), (Cl) and (Ch). The “tilde” collections are the same as the “hat” collections, since the group satisfies (Cl) and all the facts mentioned therein are applicable in this case, too. Next we discuss the vertical lines. If we assume (Cl) the two vertical dashed lines from the Sp column follows from Theorem 3.1 and from the fact that Sp = Sp in this case. If we assume (Ch) then Bp = Bp and the upper vertical dashed line in the column Bp follows from [GrS04, Theorem 1.1]. The upper dashed line in the Sp column, under the assumption (Ch) follows now by composing the corresponding existing adjacent lines. For the dotted upper vertical line in column: let C = A2(D8) and the subgroup H = Z4. Note that C≥H is empty but CH is contractible. Let now C = B2(D8) and H = Z4. In this case C≤CG(H) is empty and CH is contractible. This explains the dotted vertical line in the right column. Next consider the lower dashed vertical line in the Sp column under the assumption (Ch). Let S be a Sylow p-subgroup of G. Denote C = Sp(G) and C′ = Sp(G). Recall that |C| and |EAC| are S- homotopy equivalent; see [GrS04, Theorem 1.1]. Also, the inclusion C′ → C induces a G-homotopy equivalence on nerves; to see this combine the result of Corollary 4.6 with the homotopy properties

• f the standard collections. In order to show that |C′| → |EAC′| is an S-homotopy equivalence,

it suffices to prove that |EAC| and |EAC′| are S-homotopy equivalent. For P ≤ S, the subposet C≤CG(P) is contractible, since Op(CG(P)) > 1. We show that C′

≤CG(P) is also contractible. We

first show that the subgroup ONP = Op(NG(P)) is distinguished. Let R ∈ Sylp(NG(P)) with R = R ∩ NG(P) for some R ∈ Sylp(G). Then ONP ≤ R and also Z(R) ≤ Z(R) ≤ CNG(P)(ONP ) ≤ ONP . It follows that Z(ONP ) ∈ C′

≤CG(P). For Q ∈ C′ ≤CG(P) the contracting homotopy Q ≤ QZ(ONP ) ≥

Z(ONP ) proves the contractibility of C′

≤CG(P). Thus the inclusion C′ ≤CG(P) → C≤CG(P) induces a

homotopy equivalence. It follows that |EAC| and |EAC′| are S-homotopy equivalent and |C′| and |EAC′| are S-homotopy equivalent, too.

14

The dashed vertical line in column Ap, under the assumption (Ch) follows now from composing the horizontal solid lines with the vertical dashed line in the lower left rectangle of the table.

• Remark 6.12 (Relations between the distinguished collections and the standard ones). Obviously,

when G has just one class of elements of order p, the distinguished and the standard collections are the same. When G is of characteristic p-type, we have Bp(G) = Bcen

p

(G). But in general, the distinguished collections are not homotopy equivalent to their standard counterparts. To see this consider the example of the sporadic simple group G =Co3 and p = 2. This group satisfies both conditions (Cl) and (M). We proved, in [MgO] that B2(G) is not homotopy equivalent to B2(G) or to Bcen

2

(G). Theorem 6.13. (a). The collection Ap(G) is centralizer and normalizer sharp. (b). The collection Bp(G) is subgroup and normalizer sharp. (c). The collection Sp(G) is subgroup, centralizer and normalizer sharp. Of course, all these colections are ample.

• Proof. In Grodal [Gr02], the sharpness properties of the collection Dp(G) of principal p-

radical subgroups are investigated. These groups are by definition both p-centric and p-radical, so that Dp(G) ⊆ Sp(G) and Dp(G) ⊆ Bp(G), by Proposition 4.3. Then Grodal’s Theorems 1.2 and 7.3 from [Gr02], imply that Sp(G) and Bp(G) are subgroup and normalizer sharp. Further Ap(G) is normalizer sharp since, by Proposition 4.8 it is G-homotopy equivalent to Sp(G). Section 8 of the paper of Dwyer [Dw98] discusses the sharpness properties of elementary abelian p-subgroups, and his Theorem 8.3 reduces the question of the centralizer sharpness to the subcollec- tion of groups contained in a fixed Sylow p-subgroup S. The conical contraction of C ∩2S is obtained using the (distinguished) central elementary abelian subgroup Z = Ω1Z(S). Denote the inclusion map by j : Z → S. Given any monomorphism i : H → G, construct the subgroup H′ = i(H) · Z, with the corresponding inclusion map i′ : H′ → G. Note that Z ⊆ Z(H′) and H′ is distinguished. Then we have the zigzag of natural transformations: (H, i) → (H′, i′) ← (Z, j) proving the centralizer sharpness of Ap(G). The centralizer sharpnesss of Sp(G) follows from the G-homotopy equivalence between EAC′ and EAC proven in Theorem 4.14 for C′ = Ap(G) and C = Sp(G). The Benson collection Ep(G) is both centralizer and normalizer sharp (the latter following from an S-homotopy equivalence between Ep(G) and EAEp and a transfer argument [GrS04], and the G and S-homotopy equivalences in Theorem 3.1 imply all of the sharpness properties for Ap(G), Bp(G) and Sp(G). Note that the quaternion group Q8 of order 8 has a periodic mod 2 cohomology, but which is not detected on the (unique) central Z2 (there are nilpotent cohomology classes in the kernel of the

15

restriction map). This implies that for the group Q8, the collection A2(Q8) is not subgroup sharp, and the collection B2(Q8) consisting only of the group Q8 is not centralizer sharp.

• Open questions.
• 1. The homotopy relation between Ep(G) and

Ap(G) when the group G is of characteristic p-type.

• 2. For C =

Sp, under the assumption (M), what is the homotopy relation between EOC, EAC and C?

• 3. The relation between

Sp and Bp when G does not satisfy any of the three hypotheses (M), (Cl)

• r (Ch).

References

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