NEW COLLECTIONS OF p-SUBGROUPS AND HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS

JOHN MAGINNIS AND SILVIA ONOFREI

Abstract. Let G be a finite group and p a prime dividing its order. We define new collections of p-subgroups of G. We study the homotopy relations among them and with the standard collections of p-subgroups. We determine their ampleness and sharpness properties.

1. Introduction

The collection of all p-subgroups of a group G was introduced by Brown [Br75]; elementary abelian p-subgroups were studied by Quillen [Qu78] and p-radical subgroups by Bouc [Bou84]. These three collections are G-equivariantly homotopy equivalent; see [ThWb91].

Dwyer [Dw97] developed the concepts of ampleness and sharpness for a collection C of psubgroups of a finite group G, building on work of Webb [Wb87], Jackowski-McClure [JM92] and Jackowski-McClure-Oliver [JMO92]. Sharp collections determine mod p homology approximations for the classifying space BG; the mod p homology of the group G is determined either by the homology of the subgroups in the collection, or by the centralizers of these subgroups or by their normalizers. Dwyer [Dw98] and Grodal [Gr02] proved sharpness results for the collection of p-centric and p-radical subgroups and for the collection of principal p-radical subgroups. A comprehensive study of the homotopy equivalences between nine standard collections, as well as their sharpness properties, is presented in Grodal and Smith [GrS04].

In this paper we investigate new collections of p-subgroups Â_{p}(G), B̂_{p}(G) and Ŝ_{p}(G), the collections of elementary abelian p-subgroups (or p-radical subgroups, or p-subgroups) consisting of those p-subgroups H such that the center Z(H) contains an element of order p of central type (it lies in the center of some Sylow p-subgroup of G). We call such subgroups distinguished. We prove that these collections have the same sharpness properties as do the usual Quillen collection A_{p}(G), the Bouc collection B_{p}(G) and the Brown collection S_{p}(G). We also investigate equivariant (either for G or for a Sylow p-subgroup of G) homotopy equivalences between these collections.

For a finite Chevalley group in characteristic p, the Bouc collection is the collection of unipotent radicals of parabolic subgroups, and it corresponds to the barycentric subdivision of the associated Tits building. In an attempt to generalize the theory of buildings for Lie groups, several p-local geometries have been constructed for the sporadic finite simple groups. Following the

Date: Monday, October 1, 2005.

1991 Mathematics Subject Classification. 20J06, 20J05, 51E30, 55P91, 55R35.

2 JOHN MAGINNIS AND SILVIA ONOFREI

pioneering work of Buekenhout, Ronan and Smith [RS80] constructed 2-local geometries based on the set of those 2-local subgroups which are also 2-constrained. For the sporadic groups, there is no uniform treatment of the p-local geometries; in fact, most of these groups have more than one such geometry. However, many sporadic groups, such as those of characteristic p-type, have a p-local geometry whose barycentric subdivision corresponds to the Bouc collection.

In Smith and Yoshiara [SY97], Sawabe [Sa03] and Benson and Smith [BS04] the p-local geometries for the sporadic groups are compared to some of the collections of p-subgroups. One focus of the work of Benson and Smith is the p-local approach to the structure of 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. They describe homotopy equivalences between most of these collections and one of the standard collections, such as the Quillen A_{2}(G), the Bouc B_{2}(G) or the collection B_{2}^{cen}(G) of 2-radical and 2-centric subgroups.

Using the work of Jackowski and McClure, Benson [Ben94] constructed a subcollection E_{p}(G) of the Quillen collection. This subcollection is both centralizer and normalizer sharp. Benson uses this subcollection for p = 2 and G =Co_{3}, in which case it is homotopy equivalent to a 2-local geometry Δ first mentioned by Ronan and Stroth [RSt84] (note one of the maximal 2-local subgroups involved is not 2-constrained). Although this geometry is not homotopy equivalent to the Bouc collection, in [MgO] we prove that Δ is homotopy equivalent to B̂_{2}(Co_{3}), the collection of 2-radical subgroups which contain a central involution lying in the center of some Sylow 2-subgroup. This result led us to the definition of distinguished collections of p-subgroups.

In Section 2 a brief review of basic results from the homotopy theory of posets and a few facts regarding the best known collections of p-subgroups are given. This section also contains basics on homology decompositions, ampleness and sharpness. In Section 3 the “tilde” collections are defined. For a collection C_{p}(G) of p-subgroups, let C̃_{p}(G) denote the subcollection of C_{p}(G) whose members contain in their centers at least one element from the set E_{0} (see Section 2 for the definition of E_{0}). Theorem 3.1 contains results on the homotopy equivalences between Ã_{p}(G), B̃_{p}(G), S̃_{p}(G) as well of their associated categories EOC̃ and EAC̃. In Section 4 the distinguished collections of p-subgroups are introduced. Theorem 4.1 contains results on the homotopy equivalences between these collections and their categories, under one of three extra assumptions, denoted (ℭ𝔥), (ℭ𝔩) and (𝔐). In Section 5 it is proved that all the collections introduced in the previous two sections are ample and that their sharpness properties are similar to those of their standard counterparts.

Acknowledgement: We would like to thank Stephen Smith for providing an early version of the manuscript of Benson and Smith [BS04], the most valuable reference for the material discussed in this paper.

COLLECTIONS OF p-SUBGROUPS 3

2. Notations, terminology and standard results

Basics and notations. Assume that G is a finite group and p a prime dividing its order. Denote by Γ_{p}(G) the family of elements of order p in G which are conjugate to a central element, that is some element of order p which lies in the center of a Sylow p-subgroup of G. A p-subgroup R of G is called p-radical if R = O_{p}(N_{G}(R)). The p-subgroup R is p-centric if Z(R) is a Sylow p-subgroup in C_{G}(R). Recall that a finite group G is called of characteristic p-type if for any p-local subgroup H the following holds: C_{H}(O_{p}(H)) ≤ O_{p}(H).

Poset homotopy. In this section we review a few standard results on the homotopy theory of posets, most of them due to Quillen [Qu78] and proved in the context of equivariant homotopy by Thévenaz and Webb [ThWb91].

Notation 2.1. Let X be a poset; set X_{>x} = {y ∈ X | y > x}. Similarily, we define X_{≥x}, X_{<x}, X_{≤x}. Also |X | will denote the associated simplicial complex.

Definition 2.2. A poset X is conically contractible if there is a poset map f : X → X and an element x_{0} ∈ X such that

x ≤ f(x) ≥ x_{0}, ∀x ∈ X  or  x ≥ f(x) ≤ x_{0}, ∀x ∈ X

Lemma 2.3. A conically contractible poset is contractible. If X is conically contractible via a map f which is also G-equivariant then X is G-contractible.

Notation 2.4. Let G be a group and H a subgroup of G. Let X be a G-poset. Then X^{H} denotes the corresponding fixed point set.

Theorem 2.5. [Ben91] Suppose that X and Y are two finite G-posets and f : |X | → |Y| is a G-map. The poset map f is a G-homotopy equivalence if and only if, for all subgroups H < G, the map f restricts to an ordinary homotopy equivalence f^{H} : |X |^{H} → |Y|^{H}.

Theorem 2.6. [ThWb91] Let G be a group, let X , Y be G-posets and let f : X → Y be a map of G-posets. Suppose that either:

(i) for all y ∈ Y, f^{−1}(Y_{≤y}) is G_{y}-contractible or (ii) for all y ∈ Y, f^{−1}(Y_{≥y}) is G_{y}-contractible.

Then f is a G-homotopy equivalence.

Remark 2.7. Note that f^{−1}(Y_{≤y}) = {x ∈ X |f(x) ≤ y}. If f is the inclusion map then f^{−1}(Y_{≤y}) = X_{≤y} with y ∈ Y.

Proposition 2.8. [ThWb91] Let Y be a G-poset and X a G-invariant subposet of Y such that for each y ∈ Y \ X, the subposet Y_{<y} (or dually Y_{>y}) is G_{y}-contractible. Then the inclusion X → Y is a G-homotopy equivalence.

Proposition 2.9. [S03] If f : X → X is a G-equivariant poset map such that f ≤ Id_{X} (or dually f ≥ Id_{X}), then X is G-homotopy equivalent to any G-invariant subposet Y which also satisfies the condition f(X) ⊆ Y ⊆ X.

Standard collections of p-subgroups.

Definition 2.10. A collection C_{p}(G) of p-subgroups of G is a set of p-subgroups which is closed under conjugation. A collection is a G-poset under the inclusion relation with G acting by conjugation. Let |C_{p}(G)| denote the corresponding simplicial complex.

Notation 2.11. Let Q ∈ G be a subgroup; then C_{p}(G)^{Q} = {P |P ∈ C_{p}(G), Q ≤ N_{G}(P)} will denote the subcollection of C_{p}(G) fixed under the action of Q.

The following collections are standard in the literature:

A_{p}(G) = {E | E nontrivial elementary abelian p-subgroup of G};
S_{p}(G) = {P | P nontrivial p-subgroup of G};
B_{p}(G) = {R | R nontrivial p-radical subgroup of G}.

The simplicial complex |A_{p}(G)| is known as the Quillen complex, |S_{p}(G)| is known as the Brown complex and |B_{p}(G)| is called the Bouc complex. The three complexes are G-homotopy equivalent [ThWb91].

Let Ce_{p}(G) denote the subcollection of S_{p}(G) consisting of p-centric subgroups and let B_{p}^{cen}(G) = Ce_{p}(G) ∩ B_{p}(G) be the collection of nontrivial p-radical and p-centric subgroups. These two collections are not in general homotopy equivalent with S_{p}(G). The inclusion map B_{p}^{cen}(G) ⊆ Ce_{p}(G) induces a G-homotopy equivalence.

The following collection, denoted E_{p}(G), was introduced by Benson [Ben94] in order to study the mod 2 cohomology of Co_{3}. Start by considering the smallest subset E_{0} of G which satisfies the following properties:

i) contains the elements of order p in the center of a Sylow p-subgroup of G; ii) is closed under conjugation in G; iii) is closed under taking products of commuting elements.

Then E_{p}(G) is the collection of elementary abelian p-subgroups of G which are subsets of E_{0}. We will call |E_{p}(G)| the Benson complex.

Homology decompositions. In what follows C will stand for a collection of subgroups of the finite group G. Let p be a prime dividing the order of G and 𝔽_{p} the finite field with p elements. Let BG be the classifying space of G and EG its universal cover. Also 𝔇 will denote a small category. Let G-𝔖𝔢𝔱 be the category of transitive G-sets, which we shall regard embedded in G-𝔖𝔭, the category of G-spaces. For a G-space X let EG ×_{G} X denote the corresponding Borel construction.

COLLECTIONS OF p-SUBGROUPS 5

Definition 2.12. A mod p homology decomposition of BG consists of a functor F̃ : 𝔇 → G-𝔖𝔢𝔱 with the property that the map

hocolim_{𝔇}(EG ×_{G} F̃) → BG

is a mod p homology isomorphism.

Denote X = hocolim_{𝔇}F̃. Let F : 𝔇 → 𝔖𝔭 be given by F = EG ×_{G} F̃. For every d ∈ Obj(𝔇), the values F (d) = EG ×_{G} F̃(d) have the same homotopy type as BG_{x}, with G_{x} the isotropy group of any element x ∈ F̃(d).

A homotopy colimit gives rise to a Bousfield-Kan homology spectral sequence with

E_{ij}^{2} = colim_{𝔇}_{i}H_{j}(F, 𝔽_{p}) ⇒ H_{i+j}(hocolim_{𝔇}F; 𝔽_{p}) ≃ H_{i+j}(BG; 𝔽_{p}).

Definition 2.13. A homology decomposition is sharp if its Bousfield-Kan homology spectral sequence collapses at the E^{2}-page onto the vertical axis. This can be rephrased as follows:

colim_{𝔇}_{i}H_{j}(EG ×_{G} F̃; 𝔽_{p}) = 0 for i > 0
colim_{0}H_{∗}(EG ×_{G} F̃; 𝔽_{p}) ≃ H_{∗}(BG; 𝔽_{p}).

From the E^{2}-page onward, the Leray spectral sequence of the map:

φ : EG ×_{G} X = hocolim_{𝔇}F → hocolim_{𝔇}(∗) ≃ |𝔇|

is the Bousfield-Kan homology spectral sequence for hocolim_{𝔇}F.

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:

X/G = (hocolim_{𝔇}F̃)/G = hocolim_{𝔇}(F̃/G) ≃ hocolim_{𝔇}(∗) = |𝔇|.

Therefore the Bousfield-Kan homology spectral sequence can be identified, from the E^{2}-page onward, with the isotropy spectral sequence associated to the action of G on X.

The E^{1}-page of the isotropy spectral sequence is

E_{ij}^{1} = H_{j}(G; 𝔽_{p}[X_{i}])

where X_{i} is the set of i-simplices of X. This can be regarded as the chain complex C_{i}^{G}(X; H) for the coefficient functor H(A) = H_{j}(G; A); for any 𝔽_{p}[G]-module A. The boundary maps are induced by the alternating sum of face maps in X. Thus the E^{2}-page is isomorphic to the Bredon homology of X. The sharpness properties can be rephrased in terms of the Bredon homology and acyclicity of the G-space X.

Details on homology decompositions and their treatment in terms of simplicial sets can be found in Dwyer [Dw97, Dw98, DwH01], Grodal [Gr02], Grodal and Smith [GrS04].

6 JOHN MAGINNIS AND SILVIA ONOFREI

Ample collections and three homology decompositions. We regard C as a category

whose objects are the subgroups H ∈ C, and with morphisms the inclusion maps. There is one morphism H → K if H ⊂ K and no other morphisms. The group G acts by conjugation and the isotropy groups are of the form N_{G}(H).

Definition 2.14. A collection C of subgroups is ample if the natural map

EG ×_{G} |C| → BG

is a mod p homology isomorphism.

Remark 2.15. In the context of the general construction of a homology decomposition, if we let 𝔇 = C and F̃ : C → G-𝔖𝔢𝔱 be the constant functor F̃(d) = ∗ for every object d in 𝔇, then hocolim_{C}F̃ = hocolim_{C}(∗) = |C|. Then the above definition says that a collection is ample when F̃ provides a homology decomposition.

Associated to an ample collection of subgroups there are three different homology decompositions: the centralizer, the normalizer and the subgroup decomposition.

The C-conjugacy category A_{C} is the category whose objects are pairs (H, γ), where H is a group and γ is a conjugacy class of monomorphisms i : H → G with i(H) ∈ C. The arrows (H, γ) → (H′, γ′) are group homomorphisms j : H → H′ such that j(γ) = γ′. Let EA_{C} denote the category whose objects are pairs (H, i) where H is a group^{1} and i : H → G is a monomorphism with i(H) ∈ C. A morphism (H, i) → (K, j) is a group homomorphism ρ : H → K with jρ = i. The action of G is given by g · (H, i) = (H, c_{g}i), where c_{g} : G → G is given by c_{g}(x) = gxg^{−1}. The isotropy groups are the form C_{G}(i(H)). The collection C is ample if and only if the map EG ×_{G} |EA_{C}| → BG induces an isomorphism in mod p homology. The decomposition associated to this map is called the centralizer decomposition.

The C-orbit category O_{C} is the category whose objects are the G-sets G/H, H ∈ G and whose morphisms are G-maps. Let EO_{C} 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 ^{x}H = xHx^{−1} with H ∈ C. The collection C is ample if and only if the map EG ×_{G} |EO_{C}| → BG induces an isomorphism in mod p homology. The decomposition associated to this map is called the subgroup decomposition.

Finally let sC be the category whose objects are nonempty proper inclusion chains σ = (H_{0} < … < H_{n}) of elements in C. There is an arrow σ → σ′ when σ′ is a subchain of σ. The G-action is induced by g · H = gHg^{−1} and the isotropy groups are of the form ∩_{i}N_{G}(H_{i}). There is a functor f : sC → C given by f(H_{0} < … < H_{n}) = H_{0} and f(σ → σ′) = (H_{0} → H_{i}). This induces a homotopy equivalence |sC| → |C| which respects the G-action.

1Note A_{C} and EA_{C} are not small categories; however, they are equivalent to small categories.

COLLECTIONS OF p-SUBGROUPS 7

Let \overline{s}C be the orbit simplex category, whose objects are G-conjugacy classes of strict chains of subgroups in C. There is a morphism \overline{σ} → \overline{σ}′ if for some simplices σ ∈ \overline{σ} and σ′ ∈ \overline{σ}′ the simplex σ′ is a face of σ. The collection C is ample if and only if the map EG ×_{G} |sC| → BG is a mod p homology isomorphism. The decomposition associated to this map is called the normalizer decomposition.

A collection C of p-subgroups is called centralizer sharp, subgroup sharp, or normalizer sharp if the corresponding homology decomposition is sharp. This is equivalent to the nerves of EA_{C}, EO_{C} or C being acyclic for Bredon homology.

Remark 2.16. The three categories EO_{C}, C and EA_{C} are equivalent and therefore they have homotopy equivalent nerves. In general these equivalences are not G-homotopy equivalences. For a subgroup H of G, the fixed point sets of the above categories are related by the equivalences EO_{C}^{H} → C_{≥H} and EA_{C}^{H} → C_{≤C_{G}(H)}.

Lemma 2.17 ([GrS04], 2.2). Let G be a finite group and C a collection of subgroups of G. Suppose that F : C → C is a G-equivariant poset endomorphism satisfying either F ≤ Id_{C} or F ≥ Id_{C}. Set C′ = F (C).

(1) If F ≥ Id_{C}, then the inclusion EO_{C′} → EO_{C} induces a G-homotopy equivalence on nerves.

(2) Assume for all P ∈ C that C_{G}(P) ≤ C_{G}(F (P)). Then the inclusion EA_{C′} → EA_{C} induces a G-homotopy equivalence on nerves.

(3) The inclusion C′ → C induces a G-homotopy equivalence on nerves.

Lemma 2.18 ([GrS04], 2.5). Let G be a finite group and C′ ⊂ C collections of subgroups of G.

(1) Assume for all P ∈ C \ C′ that C_{>P} is contractible. Then |EO_{C′}| → |EO_{C}| is a Ghomotopy equivalence.

(2) Assume for all P ∈ C \ C′ that C_{<P} is contractible. Then |EA_{C′}| → |EA_{C}| is a Ghomotopy equivalence.

(3) Assume either that C_{>P} is N_{G}(P)-contractible for all P ∈ C \ C′, or that C_{<P} is N_{G}(P)- contractible for all P ∈ C \ C′. Then |C′| → |C| is a G-homotopy equivalence.

For a comprehensive analysis of the ampleness and sharpness properties of the collections of p-subgroups mentioned in the previous section; see Grodal and Smith [GrS04]. We give a brief overview of these properties here. All five collections mentioned above are ample. The collection S_{p}(G) is subgroup, centralizer and normalizer sharp, the collection A_{p}(G) is normalizer and centralizer sharp and B_{p}(G) is subgroup and normalizer sharp. Both collections Ce_{p}(G) and B_{p}^{cen}(G) are subgroup and normalizer sharp. The collection E_{p}(G) is normalizer and centralizer sharp.

8 JOHN MAGINNIS AND SILVIA ONOFREI

3. Collections related to the Benson collection

Let C_{p}(G) be any collections of p-subgroups of G. For a p-subgroup P of G define:

P̃ = Ω_{1}Z(P) ∩ E_{0}

Note this is a group since E_{0} is closed under commuting products. Further, denote:

C̃_{p}(G) = {P |P ∈ C_{p}(G) and P̃ ≠ 1} ⊆ C_{p}(G)

Theorem 3.1. Let C be one of the collections E_{p}(G), Ã_{p}(G), S̃_{p}(G) or B̃_{p}(G). For simplicity we will omit the group G. Then there exist certain homotopy equivalences, summarized in the following table:

E_{p}  Ã_{p}  S̃_{p}  B̃_{p} |EO_{C}|  • · · · · · · · · · · · · • · · · · · · · · · · · · •  • |C|  •⋮⋮⋮  •⋮⋮⋮  •||  •|| |EA_{C}|  •||  •||  •|| · · · · · · · · · · · · •⋮⋮⋮

Notation 3.2. A solid line corresponds to a G-homotopy equivalence, a dashed line to an Shomotopy equivalence and a dotted line to an ordinary homotopy equivalence. Here S denotes a Sylow p-subgroup of G.

Proposition 3.3. The inclusion i : E_{p}(G) → Ã_{p}(G) induces a G-homotopy equivalence on nerves.

Proof. Let P ∈ Ã_{p}(G), then:

E_{p}(G)_{≤P} = {E ∈ E_{p}(G)|E ≤ P } = Ã_{p}(P̃) = A_{p}(P̃)

is contractible since it is a cone on P̃. The poset E_{p}(G)_{≤P} is also N_{G}(P)-contractible and the conclusion follows by an application of Theorem 2.6(i). □

Proposition 3.4. The inclusion map i : Ã_{p}(G) → S̃_{p}(G) induces a G-homotopy equivalence on nerves.

Proof. We show that Ã_{p}(G)_{≤P} is contractible for any P ∈ S̃_{p}(G) and we apply Theorem 2.6(i). If Q ∈ Ã_{p}(G)_{≤P} then QP̃ ∈ Ã_{p}(G)_{≤P}. We obtain the following double inequality Q ≤ QP̃ ≥ P̃ which provides a contracting homotopy. The N_{G}(P)-contractibility follows from the fact that the two inequalities correspond to poset maps which are N_{G}(P)-equivariant. □

COLLECTIONS OF p-SUBGROUPS 9

Lemmatta 3.5. Let P ∈ S̃_{p}(G) and H ∈ S_{p}(G) be such that P ≤ H ≤ N_{G}(P). Then H ∈ S̃_{p}(G).

Proof. Since P is normal in N_{G}(P), it follows that P is also normal in H. Further, Ω_{1}Z(P) is normal in H. Since P̃ = E_{0} ∩Ω_{1}Z(P) it follows that P̃ is also normal in H. Then P̃∩Z(H) ≠ 1; thus H̃ ≠ 1. □

Proposition 3.6. The inclusion i : B̃_{p}(G) → S̃_{p}(G) induces a G-homotopy equivalence on nerves.

Proof. Assume that P ∈ S̃_{p}(G) \ B̃_{p}(G) and Q ∈ S̃_{p}(G)_{>P}. Then P < N_{Q}(P) ≤ N_{G}(P) and also P < O_{p}(N_{G}(P)) ≤ N_{G}(P). According to Lemmatta 3.5 it follows N_{Q}(P), O_{p}(N_{G}(P)) and N_{Q}(P)O_{p}(N_{G}(P)) are all in S̃_{p}(G). Therefore we obtain the chain:

Q ≥ N_{Q}(P) ≤ N_{Q}(P)O_{p}(N_{G}(P)) ≥ O_{p}(N_{G}(P))

of poset maps S̃_{p}(G)_{>P} → S̃_{p}(G)_{>P} which proves contractibility. Since these poset maps are equivariant it follows that S_{p}(G)_{>P} is N_{G}(P)-contractible. Finally apply Proposition 2.8. □

Proof of Theorem 3.1. The solid horizontal lines from the center row are given by the Propositions 3.3, 3.4 and 3.6. For the first solid horizontal line on the bottom row, consider P ∈ Ã_{p}(G)\ E_{p}(G) and note that Ã_{p}(G)_{<P} is conically contractible via Q ≥ Q̃ ≤ P̃, with P > Q ∈ Ã_{p}(G). Then apply the result of Lemma 2.18(2). For the solid line on the top row, combine the proof of Proposition 3.6, in particular the fact that for P ∈ S̃_{p}(G) \ B̃_{p}(G), the poset S̃_{p}(G)_{>P} is contractible, with Lemma 2.18(1).

Next we prove the G-homotopy equivalence corresponding to the middle solid horizontal line on the bottom row. We let C = S̃_{p}(G) and C′ = Ã_{p}(G). Let H be a subgroup of G. We show that EA_{C′}^{H} → EA_{C}^{H} induces a homotopy equivalence on nerves. Recall from Remark 2.16 that there is an equivalence of categories EA_{C} → C_{≤C_{G}(H)}. To simplify the notation we let X = C′_{≤C_{G}(H)} and Y = C_{≤C_{G}(H)}. We claim that the inclusion i : X → Y induces a homotopy equivalence on nerves. The proof uses a nonequivariant version of Theorem 2.6(i); we show that X_{≤P} is contractible for any P ∈ Y. First note P̃ ∈ X. If Q ∈ X_{≤P} then QP̃ ∈ X_{≤P} also. The contractibility follows from the double inequality Q ≤ QP̃ ≥ P̃ which corresponds to poset maps.

The dashed vertical line in the E_{p}(G) column, follows from [GrS04, Theorem 1.1]. The dashed vertical line from the columns corresponding to Ã_{p}(G) and S̃_{p}(G) are obtained by combining the parallel vertical dashed line with the adjacent solid horizontal lines.

Let S be a fixed Sylow p-subgroup of G. We show that, for any subgroup H of S and C = S̃_{p}(G), the functor EO_{C}^{H} → C^{H} induces a homotopy equivalence on nerves. First recall that EO_{C}^{H} is equivalent to C_{≥H}; see Remark 2.16. Therefore we have to show that the inclusion C_{≥H} → C^{H} induces a homotopy equivalence on nerves. Let Q ∈ C^{H}, thus Q ∈ S̃_{p}(G) and H ≤ N_{G}(Q).

10 JOHN MAGINNIS AND SILVIA ONOFREI

Next note that Q ≤ QH ≤ N_{G}(Q) and by Lemmatta 3.5, we have QH ∈ S̃_{p}(G). Also QH ∈ C^{H} since H normalizes QH. It follows that the map Q → QH is a poset endomorphism on C^{H} with image in C_{≥H}. The top dashed line from column S̃_{p}(G) follows now. The dashed line in the B̃_{p}(G) column is obtained by composing the solid horizontal lines and the dashed line from the upper right corner rectangle.

Next we consider an example which shows that the three dotted horizontal lines cannot be replaced by dashed (or solid) lines. Let G = D_{8}. This group contains a unique central involution and four involutions of noncentral type. There are two subgroups of the form ℤ/2 × ℤ/2 and a subgroup ℤ/4. Therefore E_{2}(G) contains a unique point and Ã_{2}(G) is formed of two edges sharing a common vertex. Let H be one of the rank two elementary abelian subgroups, then E_{2}(G)_{≥H} is empty while Ã_{2}(G)_{≥H} is a point. This shows the dotted line on the left of the top row in our table cannot be replaced by a dashed line. Next, note that C_{G}(H) = H and thus S̃_{2}(G)_{≤C_{G}(H)} is an edge, but B̃_{2}(G)_{≤C_{G}(H)} is empty. This takes care of the dotted line in the right bottom row. Finally, let K = ℤ/4. Then Ã_{2}(G)_{≥K} is empty, while S̃_{2}(G)_{≥K} contains an edge, thus it is contractible. This explains the middle dotted line on the top row. □

4. Distinguished collections of p-subgroups

In this section we define new collections of p-subgroups, we study some of their homotopy properties and the homotopy relations among them and with the standard collections.

Definitions and basic properties. Recall that Γ_{p}(G) stands for the family of elements of order p of central type in G. For a p-subgroup P of G define:

P̂ = ⟨x|x ∈ Ω_{1}Z(P) ∩ Γ_{p}(G)⟩

Further, for C_{p}(G) a collection of p-subgroups of G denote:

Ĉ_{p}(G) = {P |P ∈ C_{p}(G) and P̂ ≠ 1 }

We call Ĉ_{p}(G) the distinguished C_{p}(G) collection. We shall refer to the subgroups in Ĉ_{p}(G) as distinguished subgroups.

Remark 4.1. If G has one conjugacy class of elements of order p, it is obvious that Ĉ_{p}(G) = C_{p}(G). Clearly if P ≤ G is a p-subgroup then P̂ ≤ P̃. Also Ĉ_{p}(G) ⊆ C̃_{p}(G).

Proposition 4.2. All the p-centric subgroups of G are distinguished, that is Ce_{p}(G) ⊆ Ŝ_{p}(G). Consequently Ce_{p}(G) = Ĉe_{p}(G) and B_{p}^{cen}(G) = B̂_{p}^{cen}(G).

Proof. Let P ∈ Ce_{p}(G). Let us assume by way of contradiction that P is not distinguished. Let S be any Sylow p-subgroup of G which contains P. Then Z(S) ≤ C_{G}(P). Since P is not distinguished it follows that Z(S) ∩ Z(P) = 1. But this implies that there exists nontrivial p-elements in C_{G}(P) \ Z(P) and contradicts the fact that P is centric. □

COLLECTIONS OF p-SUBGROUPS 11

Corollary 4.3. The collection of distinguished p-radical subgroups contains the collection of p-centric and p-radical subgroups: B_{p}^{cen}(G) ⊆ B̂_{p}(G).

Proposition 4.4. For a finite group G of characteristic p-type, the following hold: (i). Every p-radical subgroup is p-centric. (ii). Every p-radical subgroup is a distinguished p-subgroup.

Proof. (i). Let R be a p-radical subgroup of G thus R = O_{p}(N_{G}(R)). Since G is of characteristic p-type, R contains the group

C_{N_{G}(R)}(O_{p}(N_{G}(R))) = C_{N_{G}(R)}(R) = C_{G}(R) ∩ N_{G}(R) = C_{G}(R)

It follows now that C_{G}(R) = Z(R) and obviously R is p-centric. (ii). Follows from part (i) and Proposition 4.3. □

Corollary 4.5. If G is a finite group of characteristic p-type, then B_{p}(G) = B̂_{p}(G) = B_{p}^{cen}(G).

Before proceeding to the study of the distinguished collections of p-subgroups we formulate the following conditions:

(𝔐) Given any P ∈ Ŝ_{p}(G); the subgroup N_{G}(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 4.6. Note that if R is a p-subgroup of G and if N_{G}(R) contains a Sylow p-subgroup of G, then R is distinguished. This is easy to see since, in this case R◁S ∈ Syl_{p}(N_{G}(R)) ⊆ Syl_{p}(G). Thus R ∩ Z(S) ≠ 1.

Remark 4.7 (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. The sporadic groups of characteristic 2-type are: M_{11}, M_{22}, M_{23}, M_{24}, J_{1}, J_{3}, J_{4}, Co_{2} and Th. The elements of 𝔖 which satisfy (𝔐) are M_{12}, M_{24}, J_{1}, J_{2}, J_{4}, McL, Suz, Co_{3}, Co_{1}, Th, M, He and O’N. The condition (ℭ𝔩) holds for the following groups in 𝔖: M_{12}, J_{2}, HS, Suz, Co_{3} and Ru. We can add to the above the following groups: M_{11}, M_{22}, M_{23}, J_{1}, J_{3}, McL, Th, O’N, Ly, Fi_{23}. The first nine groups in this last set contain one class of central involutions, while Fi_{23} has three classes of involutions, all central. In these last 10 cases, the distinguished collections are the same as the standard ones. The groups which do not satisfy any of the three conditions are Fi_{22}, Fi′_{24}, HN and BM. However, in all these four cases B̂_{2}(G) = B_{2}^{cen}(G).

12 JOHN MAGINNIS AND SILVIA ONOFREI

Some homotopy equivalences. In the sequel we will discuss homotopy properties of the distinguished complexes, in comparison with similar properties of their standard counterparts. Afterwards, we analyze the homotopy relations between the distinguished collections under one of the three above mentioned conditions. We also analyze the homotopy relations between the corresponding categories EO_{C} and EA_{C}.

Proposition 4.8. The inclusion Â_{p}(G) → Ŝ_{p}(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 ∈ Ŝ_{p}(G) and Q ∈ Â_{p}(G)_{≤P} the subposet Â_{p}(G)_{≤P} is N_{G}(P)-contractible via the double inequality Q ≤ QP̂ ≥ P̂ given by N_{G}(P)-equivariant poset maps. The G-homotopy equivalence follows by an application of Theorem 2.6(ii). □

Definition 4.9. Let P ∈ S_{p}(G). Consider the subposet of distinguished p-subgroups defined by N_{>P} = {Q ∈ Ŝ_{p}(G) | P < Q ≤ N_{G}(P)}.

Lemmatta 4.10. Let P ∈ S_{p}(G) and Q ∈ Ŝ_{p}(G)_{>P}. Then N_{Q}(P) is a distinguished p-subgroup in G.

Proof. Let P and Q be as in the hypothesis. Then P < N_{Q}(P) ≤ Q. Also Z(Q) ≤ Z(N_{Q}(P)). Therefore N_{Q}(P) ∈ N_{>P} ⊆ Ŝ_{p}(G)_{>P}. □

Lemma 4.11. For P ∈ S_{p}(G), the poset Ŝ_{p}(G)_{>P} is N_{G}(P)-homotopy equivalent to the subposet N_{>P}.

Proof. Consider the poset map f : Ŝ_{p}(G)_{>P} → Ŝ_{p}(G)_{>P} defined by f(Q) = N_{Q}(P); note that f ≤ Id_{Ŝ_{p}(G)_{>P}} and that f(Ŝ_{p}(G)_{>P}) = N_{>P}. By Proposition 2.9 it follows that Ŝ_{p}(G)_{>P} is homotopy equivalent to f(Ŝ_{p}(G)_{>P}). Since the map f is N_{G}(P)-equivariant this is a N_{G}(P)- homotopy equivalence. □

Proposition 4.12. Assume that G is a finite group with the property that one of the conditions (𝔐), (ℭ𝔩) or (ℭ𝔥) holds. If P ∈ Ŝ_{p}(G) is such that P < O_{p}(N_{G}(P)) then N_{>P} is N_{G}(P)- contractible.

Proof. Let P ∈ Ŝ_{p}(G) \ B̂_{p}(G) and denote O_{NP} = O_{p}(N_{G}(P)). Further, let Q ∈ N_{>P} so P < Q ≤ N_{G}(P). For S̄ ∈ Syl_{p}(N_{G}(P)) with the property that Q ≤ S̄, let S denote a Sylow p-subgroup of G such that S̄ = S ∩ N_{G}(P). Note that since Z(S) ≤ C_{G}(P) ≤ N_{G}(P) it follows that Z(S) ≤ Z(S̄).

(𝔐). Let M a p-local subgroup of G with N_{G}(P) ≤ M and such that |M|_{p} = |G|_{p}, that is the subgroup M has the same Sylow p-subgroup as G. Denote R = O_{p}(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 ≤ QN_{R}(P)O_{NP} ≥ N_{R}(P)O_{NP}

COLLECTIONS OF p-SUBGROUPS 13

Note Q normalizes N_{R}(P) since R ⊴ M ≥ Q and Q ≤ N_{G}(P). Also Q normalizes O_{NP} ⊴ N_{G}(P) ≥ Q. From the fact that 1 ≠ Z(S) ∩ R ≤ Z(R) ≤ Z(N_{R}(P)O_{NP}) it follows that N_{R}(P)O_{NP} is distinguished. Since 1 ≠ Z(S) ∩ Z(R) ≤ Z(QN_{R}(P)O_{NP}) it follows that this last p-subgroup is also distinguished. Thus since all the inequalities correspond to N_{G}(P)- equivariant poset maps N_{>P} → N_{>P} it follows that N_{>P} is N_{G}(P)-contractible.

(ℭ𝔥). Since G is of characteristic p-type C_{N_{G}(P)}(O_{NP}) ≤ O_{NP}. Thus P < O_{NP} ≤ S̄ ≤ S so Z(S) ≤ C_{N_{G}(P)}(O_{NP}) ≤ O_{NP}. Therefore Z(S) ≤ Z(O_{NP}). It follows that O_{NP} ∈ Ŝ_{p}(G). Next, note that QO_{NP} is distinguished, which follows from the fact Z(S̄) ≤ Z(QO_{NP}). Now consider the string of poset maps N_{>P} → N_{>P}:

Q ≤ QO_{NP} ≥ O_{NP}

which proves the N_{G}(P)-contractibility of N_{>P}.

(ℭ𝔩). Consider the string of N_{G}(P)-equivariant poset maps:

Q ≤ QO_{NP} ≥ O_{NP}

In order to finish the proof we have to prove that O_{NP} ∈ Ŝ_{p}(G). Note that P̂ ◁ O_{NP} so P̂ ∩ Z(O_{NP}) ≠ 1 and using our assumption (ℭ𝔩) we obtain that Z(O_{NP}) ∩ Γ_{p} ≠ 1. □

Corollary 4.13. Assume that G satisfies one of the conditions(ℭ𝔩), (ℭ𝔥) or (𝔐). Then Ŝ_{p}(G) and B̂_{p}(G) are G-homotopy equivalent.

Theorem 4.14. Let C be one of the collections Â_{p}(G), Ŝ_{p}(G) or B̂_{p}(G). For simplicity we will omit the group G. Then there exist homotopy equivalences, summarized in the following table:

Â_{p}  Ŝ_{p}  B̂_{p} |EO_{C}|  • · · · · · · · · · · · · · · · · · ·• _{(ℭ𝔩, ℭ𝔥, 𝔐)}  • |C|  •⋮⋮⋮  •||_{(ℭ𝔩, ℭ𝔥)}_{(ℭ𝔩, ℭ𝔥, 𝔐)}  •||_{(ℭ𝔩, ℭ𝔥)} |EA_{C}|  •||_{(ℭ𝔩, ℭ𝔥)}  •||_{(ℭ𝔩, ℭ𝔥)} · · · · · · · · · · · · · · · · · ·•⋮⋮⋮

Notation 4.15. 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).

14 JOHN MAGINNIS AND SILVIA ONOFREI

Proof of Theorem 4.14. We first show how to obtain the \underline{horizontal lines}. Let C′ = Â_{p}(G) and C = Ŝ_{p}(G). The solid horizontal line on the middle row follows from Proposition 4.8. The proof for the solid horizontal line between |EA_{C′}| and |EA_{C}| is similar to that in Theorem 3.1. Let H be a subgroup of G and denote X = C′_{≤C_{G}(H)} and Y = C_{≤C_{G}(H)}. We show that X_{≤P} is C_{G}(P)- contractible for any P ∈ Y. For Q ∈ X_{≤P} we obtain the contracting homotopy Q ≤ P̂Q ≥ P̂ which proves the conical contractibility of X_{≤P}, via poset maps which are C_{G}(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 |EA_{C}| and |EA_{C′}|.

Next consider C′ = B̂_{p}(G) and C = Ŝ_{p}(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 N_{G}(P)-contractible. Next we apply Lemma 2.18(i) to obtain the G-homotopy equivalence between |EO_{C′}| and |EO_{C}|.

To see that the two remaining dotted horizontal lines cannot be replaced by dashed lines, consider the example G = D_{8}, 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 \underline{vertical lines}. If we assume (ℭ𝔩) the two vertical dashed lines from the Ŝ_{p} column follows from Theorem 3.1 and from the fact that S̃_{p} = Ŝ_{p} in this case. If we assume (ℭ𝔥) then B̂_{p} = B_{p} and the upper vertical dashed line in the column B̂_{p} follows from [GrS04, Theorem 1.1]. The upper dashed line in the S̃_{p} column, under the assumption (ℭ𝔥) follows now by composing the corresponding existing adjacent lines.

For the dotted upper vertical line in column: let C = Â_{2}(D_{8}) and the subgroup H = ℤ_{4}. Note that C_{≥H} is empty but C^{H} is contractible. Let now C = B̂_{2}(D_{8}) and H = ℤ_{4}. In this case C_{≤C_{G}(H)} is empty and C^{H} is contractible. This explains the dotted vertical line in the right column.

Next consider the lower dashed vertical line in the Ŝ_{p} column under the assumption (ℭ𝔥). Let S be a Sylow p-subgroup of G. Denote C = S_{p}(G) and C′ = S̃_{p}(G). Recall that |C| and |EA_{C}| 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 of the standard collections. In order to show that |C′| → |EA_{C′}| is an S-homotopy equivalence, it suffices to prove that |EA_{C}| and |EA_{C′}| are S-homotopy equivalent. For P ≤ S, the subposet C_{≤C_{G}(P)} is contractible, since O_{p}(C_{G}(P)) > 1. We show that C′_{≤C_{G}(P)} is also contractible. We first show that the subgroup O_{NP} = O_{p}(N_{G}(P)) is distinguished. Let \overline{R} ∈ Syl_{p}(N_{G}(P)) with \overline{R} = R ∩ N_{G}(P) for some R ∈ Syl_{p}(G). Then O_{NP} ≤ \overline{R} and also Z(R) ≤ Z(\overline{R}) ≤ C_{N_{G}(P)}(O_{NP}) ≤ O_{NP}. It follows that Z(O_{NP}) ∈ C′_{≤C_{G}(P)}. For Q ∈ C′_{≤C_{G}(P)} the contracting homotopy Q ≤ QZ(O_{NP}) ≥ Z(O_{NP}) proves the contractibility of C′_{≤C_{G}(P)}.

COLLECTIONS OF p-SUBGROUPS 15

Thus the inclusion C′_{≤C_{G}(P)} → C_{≤C_{G}(P)} induces a homotopy equivalence. It follows that |EA_{C}| and |EA_{C′}| are S-homotopy equivalent and |C′| and |EA_{C′}| are S-homotopy equivalent, too.

The dashed vertical line in column Â_{p}, 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 4.16 (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 B̂_{p}(G) = B_{p}^{cen}(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 =Co_{3} and p = 2. This group satisfies both conditions (ℭ𝔩) and (𝔐). We proved, in [MgO] that B̂_{2}(G) is not homotopy equivalent to B_{2}(G) or to B_{2}^{cen}(G).

Remark 4.17 (Relations between the Benson collection and the distinguished collection of elementary abelian subgroups). Let C′ = E_{p}(G) and C = Â_{p}(G). Assume that G satisfies (ℭ𝔩). The G-homotopy equivalence between C′ and C follows from Theorem 3.1 in which we let P̂ = P̃. Next consider the relationship between |EA_{C′}| and |EA_{C}|. Let the poset map F : C → C given by F (P) = P̂. Note that F (C) ⊆ C′ and that F ≤ Id_{C}. Also P̂ ≤ P implies that C_{G}(P) ≤ C_{G}(P̂). Thus, according to Lemma 2.17(2), the two nerves are G-homotopy equivalent. In general, the poset E_{p}(G) fails to be homotopy equivalent to Â_{p}(G). Consider G = Co_{1} and p = 2. This group satisfies (𝔐). In this case E_{2}(G) = A_{2}(G) and B̂_{2}(G) = B_{2}^{cen}(G) are not G-homotopy equivalent. The poset B_{2}^{cen}(G) has reduced Euler characteristic −2^{18} · 23^{2} · 751 which is not divisible by |G|_{2} = 2^{21}; see [BS04, Section 6.15].

5. Ampleness and sharpness properties

The following table summarizes the sharpness results for the collections defined in the previous two sections:

Ã_{p}  B̃_{p}  S̃_{p}  Â_{p}  B̂_{p}  Ŝ_{p}
s  n  y  y  n  y  y
n  y  y  y  y  y  y
c  y  n  y  y  n  y

Theorem 5.1. (a). The collections Ã_{p}(G) and Â_{p}(G) are centralizer and normalizer sharp. (b). The collections B̃_{p}(G) and B̂_{p}(G) are subgroup and normalizer sharp. (c). The collections S̃_{p}(G) and Ŝ_{p}(G) are subgroup, centralizer and normalizer sharp. Of course, all these colections are ample.

Proof of Theorem 5.1. In Grodal [Gr02], the sharpness properties of the collection D_{p}(G) of principal p-radical subgroups are investigated. These groups are by definition both p-centric

16 JOHN MAGINNIS AND SILVIA ONOFREI

and p-radical, so that D_{p}(G) ⊆ Ŝ_{p}(G) and D_{p}(G) ⊆ B̂_{p}(G), by Proposition 4.3. Then Grodal’s Theorems 1.2 and 7.3 from [Gr02], imply that Ŝ_{p}(G) and B̂_{p}(G) are subgroup and normalizer sharp. Further Â_{p}(G) is normalizer sharp since, by Proposition 4.8 it is G-homotopy equivalent to Ŝ_{p}(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 subcollection of groups contained in a fixed Sylow p-subgroup S. The conical contraction of C ∩ 2^{S} is obtained using the (distinguished) central elementary abelian subgroup Z = Ω_{1}Z(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 Â_{p}(G).

The centralizer sharpnesss of Ŝ_{p}(G) follows from the G-homotopy equivalence between EA_{C′} and EA_{C} proven in Theorem 4.14 for C′ = Â_{p}(G) and C = Ŝ_{p}(G).

The Benson collection E_{p}(G) is both centralizer and normalizer sharp (the latter following from an S-homotopy equivalence between E_{p}(G) and EA_{E_{p}} and a transfer argument [GrS04], and the G and S-homotopy equivalences in Theorem 3.1 imply all of the sharpness properties for Ã_{p}(G), B̃_{p}(G) and S̃_{p}(G).

Note that the quaternion group Q_{8} of order 8 has a periodic mod 2 cohomology, but which is not detected on the (unique) central ℤ_{2} (there are nilpotent cohomology classes in the kernel of the restriction map). This implies that for the group Q_{8}, the collection Â_{2}(Q_{8}) is not subgroup sharp, and the collection B̂_{2}(Q_{8}) consisting only of the group Q_{8} is not centralizer sharp. □

6. Appendix

In the sequel we give a few basic properties for the distinguished collections, which were not included in the main part of the paper for the reason that they were not relevant for the proofs of the theorems.

Lemma 6.1. Assume that O_{p}(G) > 1. Then the following hold:

(i). The poset Ŝ_{p}(G)_{≥O_{p}(G)} is G-contractible. Further, the two posets Ŝ_{p}(G) and Ŝ_{p}(G)_{≥O_{p}(G)}

are G-homotopy equivalent. (ii). The poset Â_{p}(G) is G-contractible. (iii). The poset B̂_{p}(G)_{≥O_{p}(G)} = B̂_{p}(G) is G-contractible.

Proof. (i). First we claim that O_{p}(G) ∈ Ŝ_{p}(G). Note O_{p}(G) ⊴ S for any S ∈ Syl_{p}(G) and so O_{p}(G) ∩ Z(S) ≠ 1; thus O_{p}(G) is distinguished. Thus Ŝ_{p}(G)_{≥O_{p}(G)} is a cone on O_{p}(G). Since

COLLECTIONS OF p-SUBGROUPS 17

O_{p}(G) is G-invariant, it follows that Ŝ_{p}(G)_{≥O_{p}(G)} is G-contractible. Consider now the product map π_{O_{p}(G)} : Ŝ_{p}(G) → Ŝ_{p}(G) defined by π_{O_{p}(G)}(P) = P O_{p}(G) for P ∈ Ŝ_{p}(G). Notice that P O_{p}(G) ∈ Ŝ_{p}(G). To see this, consider an element z ∈ P̂ such that z ∈ Z(S) for some S ∈ Syl_{p}(G). Then O_{p}(G) ≤ S and so z ∈ C_{G}(O_{p}(G)) and z ∈ Z(P O_{p}(G)). Therefore Z(P)O_{p}(G) ∈ Ŝ_{p}(G). Also note that π_{O_{p}(G)} ≥ Id_{Ŝ_{p}(G)} and that

π_{O_{p}(G)}(Ŝ_{p}(G)) = Ŝ_{p}(G)_{≥O_{p}(G)}

Now the hypotheses from Proposition 2.9 are all in place and the G-homotopy equivalence follows. (ii). This follows from part (i) and Proposition 4.8. (iii). In this case, the poset B̂_{p}(G) has a minimal element O_{p}(G). Note if R is p-radical, then O_{p}(G) ≤ R; see Smith and Yoshiara [SY97, Lemma 1.9(1)]. □

Lemma 6.2. Let G be a p-group which is not elementary abelian. Then Ŝ_{p}(G)_{<G} is conically contractible.

Proof. Assume that G is a p-group which is not elementary abelian. Thus Φ(G), the Frattini subgroup of G is a nontrivial, proper subgroup of G. It is standard that Φ(G) ◁ G. Thus Φ(G) ∩ Z(G) ≠ 1 and Φ(G) ∈ Ŝ_{p}(G). For a subgroup Q ∈ Ŝ_{p}(G), the product Q Φ(G) is also a distinguished proper p-subgroup of G. To see this, recall that 1 ≠ Φ(G) ∩ Z(G) ⊆ Z(Q Φ(G)). Therefore the map π_{Φ(G)}(Q) = Q Φ(G) is a poset map on Ŝ_{p}(G). We have the following inequalities Q ≤ Q Φ(G) ≥ Φ(G) which prove the conical contractibility. □

Lemma 6.3. Let G be a p-group. The poset Â_{p}(G) is conically contractible.

Proof. Let Z = Ω_{1}Z(G) ∈ Â_{p}(G). For Q ∈ Â_{p}(G) we have QZ ∈ Â_{p}(G) and the string of inequalities Q ≤ QZ ≥ Z which proves the conical contractibility. □

Remark 6.4. It is standard that

A_{p}(G) = {P ∈ S_{p}(G) | S_{p}(G)_{<P} is not contractible }

However, this is not true anymore for the case of distinguished complexes. To see this consider a group G such as M_{12} or Co_{3} which has two classes of involutions A and B, with A of central type and B of non-central type. Recall that, in both of the above cases, the central involutions are closed under taking products of commuting elements. Then let P be a subgroup of G such that P is of type A^{1}B^{2}. Note that S_{2}(G)_{<P} is not contractible since it consists of three vertices and no edges, the three subgroups of order 2. However, the distinguished complex Ŝ_{2}(G)_{<P} is contractible since it contains a single element, the 2-subgroup generated by the central involution in G.

Remark 6.5. In general, if Q is not special, then Ŝ_{p}(G)^{Q} is not usually contractible. Take G = M_{12} and p = 2. This group satisfies the conditions (ℭ𝔩) and (𝔐) and the collection Ŝ_{p}(G) is G-homotopy equivalent to the 2-local geometry Δ; for details see [BS04, Section 6.2]. Let z be a non-central involution in G. The fixed point set Δ^{z} is non-contractible; see [M90]. An application of Theorem 2.5 shows that Ŝ_{2}(G)^{z} is not contractible either. If we assume (ℭ𝔥) then Ŝ_{p}(G) is G-homotopy equivalent to S_{p}(G). The contractibility of the fixed point sets follows from an application of Theorem 2.5 and Quillen’s result that S_{p}(G)^{z} is contractible. In this case, Â_{p}(G) and B̂_{p}(G) are G-homotopy equivalent to Ŝ_{p}(G), thus we have similar conclusions for Â_{p}(G)^{Q} and B̂_{p}(G)^{Q}.

7. Open questions

Remark 7.1 (The relation between E_{p} and Â_{p}). In general, E_{p}(G) and Â_{p}(G) need not be homotopy equivalent (as in Co_{1}, which satisfies condition (𝔐)). However, the inclusion E_{p} ∩ Â_{p}(G) ⊂ Â_{p}(G) is an equivalence (use the poset map f(P) = P̂). It is not clear what happens when the group is of characteristic p-type.

Remark 7.2. For C = Ŝ_{p}, under the assumption (𝔐), what is the homotopy relation between EO_{C}, EA_{C} and C?

Remark 7.3 (The relation between Ŝ_{p} and B̂_{p}). When G does not satisfy any of the three hypotheses (𝔐), (ℭ𝔩) or (ℭ𝔥) we expect Ŝ_{p}(G) and B̂_{p}(G) not to be homotopy equivalent anymore. We do not have an example in which this homotopy equivalence fails.

References

[Ben91] D.J. Benson, Representations and cohomology II, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1991).

[Ben94] D.J. Benson, Conway’s group Co_{3} and the Dickson invariants, Manuscripta Math. 85(1994), no. 2, 177-193.

[BS04] D.J. Benson, S.D. Smith, Homotopy decompositions and 2-completed classifying spaces for sporadic simple groups, preprint 2004.

[Br75] K.S. Brown, Euler characteristics of groups: the p-fractional part, Invent. Math. 29 (1975), 1-5.

[Bou84] S. Bouc, Homologie de certains ensembles ordonées, Comptes Rendus Acad. Sci. Paris, Série I 299 (1984), 49-52.

[Dw97] W.G. Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36(1997), no. 4, 783-404.

[Dw98] W.G. Dwyer, Sharp homology decompositions for classifying spaces of finite groups, Group representations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Sympos. Pure. Math., vol. 63, Amer. Math. Soc., Providence, RI (1998) 197-220.

[DwH01] W.G. Dwyer, H-W. Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics - CRM Barcelona, Birkhauser (2001).

[Gr02] J. Grodal, Higher limits via subgroup complexes, Ann. of Math., 155 (2002) 405-457.

[GrS04] J. Grodal, S.D. Smith, Propagating sharp group homology decompositions, arXiv:math.AT/0402270v1 (2004).

[M90] J.S. Maginnis, Local control of group cohomology, J. Pure Appl. Algebra 103 (1995), no. 1, 81-90.

[JM92] S. Jackowski, J.E. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992) 112-132.

COLLECTIONS OF p-SUBGROUPS 19

[JMO92] S. Jackowski, J.E. McClure, R. Oliver, Homotopy classification of self-maps of BG via G-action, I, Ann. of Math. 135 (1992) 183-226.

[MgO] J.S. Maginnis, S.E. Onofrei, On a homotopy relation between the 2-local geometry and the Bouc complex for the sporadic group Co_{3} (preprint 2004).

[Qu78] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978) 101-128.

[RS80] M.A. Ronan, S.D. Smith, 2-local geometries for some sporadic groups, The Santa Cruz Conference on Finite Groups (Univ. of California, Santa Cruz, California 1979), (Providence R.I.), Proc. Symp. Pure Math., vol. 37, American Math. Society, 1980, pp. 283-289.

[RSt84] M.A. Ronan, G. Stroth, Minimal parabolic geometries for the sporadic groups, European J. Combin. 5 (1984), no.1, 59-91.

[Sa03] M. Sawabe, On a p-local geometry associated with a complex of non-trivial p-subgroups, Bull. London Math. Soc. 35 (2003) 196-202.

[SY97] S.D. Smith, S. Yoshiara, Some homotopy equivalences for sporadic geometries, J. Algebra 192 (1997) 326-379.

[S03] S.D. Smith, Subgroup complexes (unpublished manuscript, June 2003).

[Th87] J. Thévenaz, Permutation representations arising from simplicial complexes, J. Comb. Th. A 46 (1987) 121-155.

[ThWb91] J. Thévenaz, P.J. Webb, Homotopy equivalence of posets with a group action, J. Comb. Th. A 56 (1991) 173-181.

[Wb87] P.J. Webb, A local method in group cohomology, Comment. Math. Helvetici 62 (1987) 135-167.

Mathematics Department, Kansas State University, 137 Cardwell Hall, Manhattan, Kansas 66506, Email: maginnis@math.ksu.edu

Mathematics Department, University of California Riverside, 222 Surge Hall, Riverside, California 92521, Email: onofrei@math.ucr.edu