HOMOLOGY DECOMPOSITIONS FOR

CLASSIFYING SPACES OF FINITE GROUPS

ASSOCIATED TO MODULAR REPRESENTATIONS

D. Notbohm

Abstract. For a prime p, a homology decomposition of a the classifying space BG of a finite group G consist of a functor F : D ⟶ spaces from a small category into the category of spaces and a map hocolim F ⟶ BG from the homotopy colimit to BG which induces an isomorphism in mod-p homology. Associated to a modular representation G ⟶ Gl(n; 𝔽_{p}) we construct a family of subgroups closed under conjugation, which gives rise to three different homology decompositions, the so called subgroup, centralizer and normalizer decomposition. For an action of G on a 𝔽_{p}- vector space V, this collection consists of the isotropy groups of all nontrivial proper subspaces of V with nontrivial p-Sylow subgroup. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

1.Introduction.

For a finite group G and a prime p, a homology decomposition or homology approximation of the classifying space BG is a a way to glue spaces together such that this new space allows a map into BG which induces an isomorphism in mod-p homology or cohomology. More explicit, it is given by a functor F : D ⟶ spaces from a small, a discrete or, even better, a finite category D into the category of spaces and a mod-p homology isomorphism

hocolim_{D} F ⟶ BG

from the homotopy colimit of F to the classifying space BG of G. In practice, the category D is very often given by a full subcategory of the orbit category or of the category of subgroups of G. Homology decompositions of classifying spaces became an important tool in the analysis of the homotopy theory of these spaces. They can be used to make explicit calculations for G or to prove general theorems via an induction of the order of G. In the latter case one needs that the values of the functor F are homotopy equivalent to classifying spaces of smaller groups or of proper subgroups of G. And this turns out to be the case for the known decomposition results. An (incomplete) list of the usefulness and of applications of homology decompositions is given in [3].

In [3], Dwyer described many different homology decomposition formulas in terms of a single invariant: an associated poset of subgroups of G (see Section

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 55R40, 20J06, 20C20.

Key words and phrases. finite group, classifying space, modular representation, homology.

2). For each of these posets he constructed three different homology decompositions, the centralizer, the subgroup and the normalizer decomposition. Some of them were already known some of them gave new decompositions [3] [4].

The goal of this paper is to produce further homology decompositions of BG which are associated to modular representations. We hope to build a bridge between modular representation theory of a finite group and the homotopy theory of the classifying space at a fixed prime. This might give the chance to produce suitable decomposition formulas for questions about the homotopy theory of classifying spaces and/or invariant theory. The method of producing these new homology decompositions were already used with success in [8] and [9], where the author constructed spaces with polynomial mod-p cohomology and proved uniqueness results for them.

Before we formulate our results we first have to recall Dwyer’s concepts of sharp homology decompositions [4].

Given a (covariant) functor F : D ⟶ spaces, D a discrete category, the standard method to calculate the mod-p homology of hocolim_{D} F is given by the Bousfield-Kan spectral sequence. It has the form

E_{2}^{i,j} ≅ lim_{\overrightarrow{D}}^{i}H_{j}(F; 𝔽_{p}) ⟹ H_{i+j}(hocolim_{D} F; 𝔽_{p})

and converges to the mod-p homology of the homotopy colimit. Here, lim_{⟶}^{i} denotes the i-th left derived functor of the functor lim_{⟶} which maps diagrams of abelian groups to the direct limit of the diagram.

If there exists a mod-p equivalence ϕ : hocolim_{D} F ⟶ BG, i.e. ϕ induces an isomorphism in mod-p homology, then this map is called a homology decomposition. We call this decomposition sharp, if

lim_{\overrightarrow{D}}^{i}H_{∗}(F; 𝔽_{p}) = \{ \begin{matrix}0 & for i ≥ 1 \\ H_{∗}(BG; 𝔽_{p}) & for i = 0 \end{matrix}

where, for i = 0, the isomorphism is induced by ϕ.

For a collection C of subgroups of G, i.e. a family which is closed under conjugation, Dwyer constructed three different decompositions, the subgroup, the centralizer and the normalizer decomposition. The first is based on full subcategory of the orbit category and has the form

b_{C} : hocolim_{O_{C}(G)} β_{C} ⟶ BG ,

the second is based on a full subcategory of the category of subgroups and has the form

a_{C} : hocolim_{A_{C}} α_{C} ⟶ BG ,

and the last one is based on the category of orbit simplices for the G-action on a complex K_{C} associated to C and has the form

d_{C} : hocolim_{\overline{s}S_{C}} δ_{C} ⟶ BG .

For details see Section 2.

1.1 Definition. We call a collection C subgroup sharp if the map b_{C} gives a sharp decomposition, centralizer sharp if a_{C} gives a sharp decomposition and normalizer sharp if d_{C} does so.

We will construct collections of subgroups associated to modular representations which are subgroup, centralizer and normalizer sharp. Given a modular representation ρ : G ⟶ GL(V), i.e. V is a vector space over 𝔽_{p} and p divides the order of G, we consider the collection I(ρ) of the isotropy groups of all nontrivial proper subspaces of V whose order is divisible by p. The analogue definition we can make for any G-set or G-space X. In this case we denote the collections by I(X).

1.2 Theorem. Let G be a finite group such that p divides the order of G.

(i) Let ρ : G ⟶ Gl(V) be a modular representation. Then, the collection I(ρ) is subgroup, centralizer and normalizer sharp. (ii) Let X be a G-space. Then, the collection I(X) is subgroup, centralizer and normalizer sharp.

To be on the sure side in part (ii), we make the convention, that the isotropy group of the empty set is G itself.

If a collection C contains the group G itself, we do not get interesting decomposition formulas. For example, in this case, the orbit category O_{C}(G) contains G/G which is a terminal object. Therefore the subgroup decomposition contains BG as a piece and is sharp by trivial reasons. Let P be a p-Sylow subgroup of G. If V^{G} ≠ V^{P} then G ∉ I(ρ). In particular, since V^{P} is always a nontrivial subspace, this is true if the representation is fixed point free. The same holds for I(X) if X^{G} ≠ X^{P}.

Usually one is interested to choose the collection as small as possible. As the proofs will show (see in Remark 4.3), we only have to take into account the isotropy groups of the fixed point sets of nontrivial p-subgroups of G.

The method of proof for Theorem 1.3 can also be used to reconstruct the Jackowski-McClure decomposition via centralizers of nontrivial elementary abelian p-subgroups [7]. Let E denote the collection of all nontrivial elementary abelian p-subgroups of G and A the collection of all nontrivial abelian p-subgroups. Let Cen(E) := {C_{G}(E)|E ∈ E} and Cen(A) := {C_{G}(A)|A ∈ A} denote the associated collection of centralizers.

1.3 Theorem. The collections Cen(E) and Cen(A) are subgroup and normalizer sharp.

Since the collection E is not subgroup sharp, one cannot expect that these collections are centralizer sharp.

The Jackowski-McClure decomposition via centralizers of elementary abelian psubgroups is described via a functor defined on the Quillen-category of all nontrivial elementary abelian p-subgroups. This is a subcategory of the category of subgroups. The subgroup decompositions associated to Cen(E) comes from a functor defined on a subcategory of the orbit category, but one can show that both give the same decomposition formulas.

The proof of the theorems is based on the following idea. Let C be a collection of subgroups and denote by P the collection of all nontrivial p-subgroups. Then we have inclusions

C ⟶ C ∪ P ⟵ P

where C ∪ P denotes the union of both collections. We will compare the decompositions associated to these three collections.

1.4 Definition. We say that C is a minimalization of P if for every P ∈ P there exists a unique minimal element H_{P} ∈ C such that P ⊂ H_{P}, i.e. if P ⊂ H and H ∈ C then H_{P} ⊂ H.

1.5 Proposition. If C is a minimalization of P then C is subgroup sharp.

Actually, this statement should be read as, if C is a minimalization of P and one of the collection is subgroup sharp, then so is the other And that is the way we will prove this theorem. Because of Proposition 1.5 the proof of subgroup sharpness for I(ρ) and I(X) boils down to the fact that both are minimalizations of P (Lemma 4.1). The analogue result of 1.5 for centralizer and normalizer sharpness we only can prove for I(ρ) and I(X).

In the case of the collections Cen(E) and Cen(A) one has to replace P by the collection P^{c} of all p-centric p-subgroups of G. A p-subgroup P ⊂ G is called pcentric if the center Z(P) ⊂ C_{G}(P) of P is a p-Sylow subgroup of the centralizer C_{G}(P). Then the same ideas apply.

As already mentioned, the subgroup decomposition for Cen(E) is closely related to the Jackowski-McClure decomposition. Actually, it turns out that the decomposition given by p-centric p-subgroups and by centralizers of elementary abelian subgroups are equivalent; i.e. if one the underlying diagrams gives a homology decomposition then so does the other. We intend to analyze this connection in detail for compact Lie groups as well as for p-compact groups in a future paper.

The paper is organized as follows: In the next section we recall material from from [3] and [4]. In particular we explain the concept of subgroup, centralizer and normalizer decompositions. In Section 3 we compare different collection with respect to there ampleness and sharpness properties. We set up the background for the proofs of the theorem. The proofs themselves are worked out in Section 4. And the last section is devoted to a discussion of some examples.

The paper is written with the convention that ”spaces” means ”simplicial sets”. And a map X ⟶ Y between spaces is a (weak) homotopy equivalence, if it induces a (weak) homotopy equivalence between the geometric realizations.

Throughout the paper p is a fixed prime, and 𝔽_{p} denotes the finite field of pelement. A map between spaces is called a mod-p or 𝔽_{p}-equivalence if it induces an isomorphism in mod-p homology.

We thank the Sonderforschungsbereich 343 ”Diskrete Strukturen in der Mathematik at the University of Bielefeld as well as the Max-Planck-Institut für Mathematik in Bonn for support when this work was done.

2. The centralizer, the subgroup and the normalizer decomposition. For each collection C of subgroups of a finite group G, Dwyer constructed three

different decomposition, the centralizer , the subgroup and the normalizer decomposition. Moreover, Dwyer identified the Bousfield-Kan spectral sequence of the decompositions with the equivariant Bredon homology of certain G-spaces. We recall these constructions. All the material in this section is taken from [3] and [4].

Let EG denote the universal cover of BG. For a G-space X we denote by X_{hG} := (X × EG)/G the Borel construction of G.

Let γ̃ be a functor from a small category D into transitive G-sets. Every G-set can be considered as a discrete G-space. Then, the definition X := hocolim_{D} γ̃ makes sense and X is a G-space. Moreover, X can be identified with the nerve of the category whose objects are pairs (x, d) such that x ∈ γ̃(d) and whose morphisms are maps d ⟶ d′ such that γ̃(d ⟶ d′)(x) = x′.

Let γ := γ̃_{hG}. The natural maps γ̃(d) ⟶ ∗ are G-equivariant and induce maps γ(d) ⟶ BG which are compatible with the morphisms in D, and therefore a map hocolim_{D} γ ⟶ BG. If this map is a mod-p equivalence, then γ provides a homology decomposition of BG.

A coefficient functor for G is a functor Φ from the category of 𝔽_{p}[G]-modules to the category of 𝔽_{p}-vector spaces which preserves arbitrary direct sums. If K is a subgroup of G, Φ|_{K} denotes the coefficient functor for K defined by Φ|_{K}(A) := Φ(𝔽_{p}[G] ⊗_{𝔽_{p}[K]} A). Examples are given by twisted homology groups of G, namely H_{∗}(G; M ⊗ −) where M is a 𝔽_{p}[G]-module. In this case , by Shapiro’s lemma, the restriction to K is isomorphic to H_{∗}(K; M ⊗ −).

If X is a G-space and Φ a coefficient functor for G, then we can construct a chain complex C_{n}^{G}(X; Φ) := Φ(𝔽_{p}[X_{n}]) where X_{n} denotes the set of n-simplices of X. The boundary map is given by the alternating sum of the face maps in X. The Bredon homology groups H_{∗}^{G}(X; Φ) are defined to be the homology groups of this chain complex. By construction, Bredon homology is a functor from the category of G-spaces to the category of abelian groups. A G-map f : X ⟶ Y is called a Φ-equivalence if it induces an isomorphism H_{∗}^{G}(f; Φ) for Bredon homology. And a G-space X is called acyclic with respect to Φ if the map X ⟶ ∗ induces a Φ-equivalence.

The E^{2}-term of the Bousfield-Kan spectral sequence of hocolim_{D} γ can be identi-

fied with the Bredon homology groups of hocolim_{D} γ̃ with coefficient functor H_{∗}(G; −)

[4; §3 and 4.4]. And hocolim_{D} γ ⟶ BG is a sharp decomposition if and only if

hocolim_{D} γ̃ is acyclic with respect to H_{∗}(G; −). This is also true if we work with

twisted coefficients, i.e. if we take a 𝔽_{p}[G]-module M and consider H_{∗}(−; M)- homology. For details see [4; §3, 4.4 and 6.2].

In the following, C is a collection of subgroups. We can associate to C a simplicial complex K_{C} [1; 6.2]. The n-simplices of K_{C} are the subsets {H_{i}} ⊂ C of cardinality n + 1 which are totally ordered by proper inclusions. The group G acts on C and, since this action respects inclusions, it also acts on K_{C}.

Before we explain the different decompositions associated to the collection C, we notice that, for every transitive G-set X considered as a discrete G-space, the Borel construction X_{hG} produces a space homotopy equivalent to BG_{x} for some x ∈ X, where G_{x} is the stabilizer group of x.

2.1 The subgroup decomposition. We denote by O_{C}(G) ⊂ O(G) the full subcategory of the orbit category whose objects are given by the homogeneous spaces G/H where H ∈ C. As usual the morphisms are given by G-equivariant maps. Considering transitive G-sets as discrete G-spaces defines a functor β̃_{C} : O_{C}(G) ⟶ G − spaces. Composing β̃_{C} with the Borel construction (−)_{hG} gives a functor

β_{C} : O_{C}(G) ⟶ spaces

whose value at G/H has the homotopy type of BH. The natural maps β_{C}(G/H) = (G/H)_{hG} ⟶ BG are compatible with the morphisms of the orbit category and induce a map

b_{C} : hocolim_{O_{C}(G)} β_{C} ⟶ BG.

This is a homology decomposition if b_{C} is a mod-p equivalence.

The space X_{C}^{β} := hocolim_{O_{C}(G)} β̃_{C} carries a G-action and is the nerve of the category

X_{C}^{β} whose objects are pairs (gH, G/H) and whose morphism are G-maps G/H ⟶ G/H′ which map gH on g′H′. The decomposition is sharp if and only if (X_{C}^{β}) is acyclic with respect to H_{∗}(G; −).

2.2 The centralizer decomposition. The C-conjugacy category A_{C} is the category in which the objects are pairs (H, Θ) where H is a group and Θ a conjugacy class of monomorphisms i : H ⟶ G with i(H) ∈ C. A morphism (H, Θ) ⟶ (H′, Θ′) is a group homomorphism j : H ⟶ H′ (actually a monomorphism) which under composition carries Θ′ into Θ. Restricting H for instance by a requiring that H is a subgroup of G or that |H| ≤ |G|, makes A_{C} into a finite category.

We define a (covariant) functor α̃_{C} : A_{C}^{op} ⟶ G − spaces by assigning to each object (H, Θ) the transitive G-set Θ. Composing α̃_{C} with the Borel construction yields a functor

α_{C} : A_{C}^{op} ⟶ spaces ,

whose value at (H, Θ) is homotopy equivalent to BC_{G}(i(H)) for some i ∈ Θ. The natural transformation from α̃_{C} to the constant functor with value a point, establishes a natural transformation from α_{C} to the constant functor whose value is BG. Hence, we get a map

a_{C} : hocolim_{A_{C}^{op}} α_{C} ⟶ BG ,

If a_{C} is a mod-p equivalence we get a decomposition formula.

The space X_{C}^{α} := hocolim_{A_{C}^{op}} α̃_{C} carries a G-action and is the nerve of the category

X_{C}^{α} whose objects are pairs (i, H) where i ∈ Θ and whose morphisms are maps ρ : H ⟶ H′ such that i′ρ = i. The centralizer decomposition is sharp if and only if X_{C}^{α} is acyclic with respect to H_{∗}(G; −).

For later purpose we need a smaller category. Let Â_{C} denote the category whose object are given by subgroups i_{H} : H ↪ G and whose morphisms are homomorphisms H ⟶ H′ equal to a conjugation by an element of G. There exists a functor Â_{C} ⟶ A_{C} defined by H ↦ (H, Θ_{H}) where Θ_{H} is the conjugacy class of the inclusion i_{H}. The values on the morphisms are obvious. This functor is an inclusion. The image is a full subcategory of A_{C} and contains an element of each isomorphism class of objects of A_{C}. Hence, centralizer sharpness can be reduced to a question about the restriction α_{C}|Â_{C}^{op} of α_{C}.

Analogously as above we can construct a category X̂_{C}^{α} whose objects are pairs (H, gC_{G}(H) and whose morphisms (H, gC_{G}(H)) ⟶ (H′, g′C_{G}(H′)) are described by elements g_{0} ∈ G such that g_{0}Hg_{0}^{−1} ⊂ H′ and such that g′g_{0}C_{G}(H) = gC_{G}(H). This category carries an G-action which comes from the G-action on the homogeneous spaces G/C_{G}(H). And again, the decomposition a_{C} : hocolim_{Â_{C}} α_{C} ⟶ BG is

sharp if and only if X̂_{C}^{α} is acyclic with respect to H_{∗}(G; −).

2.3 The normalizer decomposition. Let \overline{s}S_{C} denote the category of the orbit simplices for the G-action on K_{C}. That is the objects are the G-orbits \overline{σ} of the simplices σ of K_{C}. There is exactly one morphism \overline{σ} ⟶ \overline{σ}′ if there exist σ ∈ \overline{σ} and σ′ ∈ \overline{σ}′ such that σ ⊂ σ′, i.e. σ is a face of σ′. There is a functor δ̃_{C} : \overline{s}S_{C} ⟶ G − spaces which assigns to each object \overline{σ} the transitive G-set \overline{σ}. Passing to the Borel construction yields a functor

δ_{C} : \overline{s}S_{C} ⟶ spaces ,

whose value at \overline{σ} is homotopy equivalent to B(⋂_{i} N_{G}(H_{i})) where (H_{0} ⊂ ... ⊂ H_{n}) ∈ \overline{σ}. The natural transformation from δ̃_{C} to the constant functor with value a point establishes a natural transformation from δ_{C} to the constant functor with value BG and therefore a map

d_{C} : hocolim_{\overline{s}S_{C}} δ_{C} ⟶ BG .

If this is a mod-p equivalence then we get a decomposition of BG.

Let sdX_{C}^{δ} := hocolim_{\overline{s}S_{C}} δ̃_{C}. And X_{C}^{δ} is the nerve of the category X_{C}^{δ} whose objects

are the subgroups H ∈ C and where there exists a unique morphism H ⟶ H′ if H ⊆ H′. Both spaces carry an G action which is induced from the G-action on C. The space sdX_{C}^{δ} is the barycentric subdivision of X_{C}^{δ}. And d_{C} is a sharp decomposition if and only if X_{C}^{δ} is acyclic with respect to H_{∗}(G; −).

All this three decomposition are closely connected as the next theorem shows. Let ∗ denote the one-point space with trivial G-action.

2.4 Definition. The collection C is called ample if the G-equivariant map K_{C} ⟶ ∗ induces a 𝔽_{p} equivalence (K_{C})_{hG} ⟶ BG.

2.5 Theorem. (Dwyer [3]) Let G be a finite group and let C be a collection. If the complex K_{C} is ample then all three maps a_{C}, b_{C} and d_{C} are give homology decompositions. And if one of this three maps is a homology decomposition then K_{C} is ample.

2.6 Remark. Let M be an 𝔽_{p}[G]-module. Then, the collection C is called M-ample if the map (K_{C})_{hG} ⟶ BG induces an isomorphisms on (twisted) homology with coefficients in M. C is called M-subgroup sharp if b_{C} induces a sharp decomposition for the twisted homology groups H_{∗}(−; M). And this is true if and only if the space X_{C}^{β} is acyclic for the coefficient functor H_{∗}(G; M ⊗ −). The same holds for the centralizer and the normalizer decompositions.

3. Comparison of collections.

Let H and K be two collections of subgroups of a finite group G. We want to show results of the following form. If K satisfies some ampleness or sharpness properties, then so does H. Of course we have to put some conditions on H. For a subgroup H ⊂ G we denote by K_{H} := {K ∈ K|K ⊂ H} the set of all elements in K which are contained in H. This is a collection of subgroups of H. Actually, we will compare H with H ∪ K and K with H ∪ K. In the latter case we like think of H ∪ K as an extension of K and in the first case of H as a reduction of H ∪ K.

3.1 Definition. Let H, K and L be three collections of subgroups of G.

(i) The collection H is called a sharp reduction of L if H ⊂ L and if for for every L ∈ L\H there exists a unique minimal element in H, denoted by H_{L}, with respect to inclusions such that L ⊂ H_{L}; i.e. if L ⊂ H and H ∈ H, then H_{L} ⊂ H. (ii) We say that L is an ample extension of K, if K ⊂ L and if K_{L} is an ample collection for all L ∈ L \ K. The collection is called a subgroup sharp extension if, in addition, K_{L} is a subgroup sharp collection for L.

3.2 Proposition. Let H ⊂ L be two collections of subgroups of a finite group G. If H is a sharp reduction of L, then the following holds:

(i) The map K_{H} ⟶ K_{L} is a homotopy equivalence. (ii) The maps X_{H}^{β} ⟶ X_{L}^{β}, X̂_{H}^{α} ⟶ X̂_{L}^{α} and X_{H}^{δ} ⟶ X_{L}^{δ} are G-equivariant homotopy equivalent. (iii) In particular H has all the ampleness and/or sharpness properties which L has.

3.3 Proposition. Let K ⊂ L be two collections of subgroups of a finite group G.

(i) If L is an ample extension of K, then the map K_{K} ⟶ K_{L} is a homotopy equivalence. (ii) If L is a subgroup sharp extension of K, then L is subgroup sharp if and only if K is.

3.4 Remark. Let M be a 𝔽_{p}[G]-module and let H, L and K as in the above propositions. In fact, Proposition 3.1 say that H has the same M-ampleness and the same M-subgroup, M-centralizer and M-normalizer sharpness properties as L. And, if K and K_{L} are M-subgroup sharp, then so is L. actually, the last claim only follows from the proof of 3.3 (ii).

Proof of Proposition 3.2. For a collection C we denote by S_{C} the nerve of the category S_{C} which is just the poset C considered as a category. Then, K_{C} ≃ S_{C}, since the simplices of K_{C} are just the nondegenerate simplices of S_{C}.

If H ⟶ L is a strong reduction we set H_{L} := L if L ∈ H. Then we can define a map F : L ⟶ L by F (L) := H_{L}.This is idempotent, respects inclusion, is Gequivariant and maps onto H. The G-equivariance follows, since gH_{L}g^{−1} = H_{gLg^{−1}}. Actually, H ⊂ L is a strong reduction if and only if such a map exists.

Basically by assumption, the under categories L ⟶ S_{H} for the inclusion S_{H} ⟶ S_{L} have an initial element L ⊂ F (L). All these under categories are contractible and the inclusion is right cofinal. By Quillens’s Theorem A [10], both categories have homotopy equivalent classifying spaces, which shows that K_{H} ≃ K_{L}.

Next we prove part (ii). The maps are induced by the G-equivariant inclusions i_{β} : X_{H}^{β} ⊂ X_{L}^{β}, i_{α} : X̂_{H}^{α} ⊂ X̂_{L}^{α} and i_{δ} : X^{δ} ⊂ X_{L}^{δ} of categories. We construct Gequivariant left inverse maps, denoted by s_{β}, s_{α} and s_{δ} and G-equivariant natural transformations id ⟶ i_{β}s_{β}, id ⟶ i_{α}s_{α} and id ⟶ i_{δ}s_{δ}.

For the subgroup decomposition, s_{β} is given by (G/L, gL) ↦ (G/F (L), gF (L)). The values on the morphisms are the obvious ones. By the properties of F, this is well defined. And the functor s_{β} is G-equivariant. The inclusion L ⊂ F (L) induces a morphisms (G/L, gL) ⟶ (G/F (L), gF (L)). The family of these morphisms establishes a natural transformation id ⟶ i_{β}s_{β} which, in addition, is G-equivariant.

For the centralizer decomposition, (L, gC_{G}(L) ↦ (F (L), gC_{G}(F (L)) establishes the left inverse. We can also define the values on the morphisms in the obvious way and get a G-equivariant functor. Again the inclusion L ⊂ F (L) establishes a morphism (L, gC_{G}(L) ⟶ (F (L), gC_{G}(F (L)) in X̂_{L}^{α}. And the family of all these morphisms establishes a G-equivariant natural transformation id ⟶ i_{α}s_{α}.

For the normalizer decomposition, the left inverse is given by L ↦ F (L) and by the obvious definition for morphisms. This is well defined and G-equivariant. Again the inclusion L ⊂ F (L) establishes a morphism L ⟶ F (L) and the family of all these morphisms a G-equivariant natural transformation id ⟶ i_{δ}s_{δ}.

The lemma below shows that the construction of the left inverse and the natural transformation is sufficient for proving Part (ii).

Part (iii) is a consequence of (ii). □

3.5 Lemma. Let i : C ⟶ D be a G-equivariant functor between small categories, which has a G-equivariant left inverse s : D ⟶ C. Let T : id ⟶ i∘s a G-equivariant natural transformation. Then, the map i induces a G-equivariant homotopy equivalence between the classifying spaces of C and D.

Proof. It is sufficient to show that the map i induces a homotopy equivalence between the fixed-point sets of the classifying spaces BC and BD of the categories for all subgroups of G. Since the (BC)^{H} ≃ B(C^{H}), passing to fixed-points maintains the situation described in the statement but where we forget the equivariant setting. We only have to prove the nonequivariant version of the statement. But that is obvious since a natural transformation between functors induces a homotopy between the induced maps on the classifying spaces.

Proof of Proposition 3.3. Let K ⟶ L be an ample extension. We want to show that K_{K} is ample if and only K_{L} is or, equivalently, that b_{K} : hocolim_{O_{K}(G)} β_{K} ⟶ BG

is a homology decomposition if and only if b_{L} is one.

For the inclusion O_{K}(G) ⟶ O_{L}(G), the left Kan-extension of β_{K} is given by L(β_{K})(G/L) := hocolim_{O_{K}⟶G/L} β_{K} and hocolim_{O_{K}(G)} β_{K} ≃ hocolim_{O_{L}(G)} L(β_{K}) (e.g. see [5; 9.8]).

There exists a natural transformation T : L(β_{K}) ⟶ β_{L} ≃ BL, since all objects as well as all morphisms in the under category are given by G–maps. There is a functor O_{K_{L}}(L) ⟶ (O_{K} ⟶ G/L) which is an equivalence of categories. The inverse is given by mapping G/K ⟶ G/L to the counter image of the orbit 1 · L ∈ G/L. By assumption, the collection K_{L} is ample which implies that the natural transformation T is an equivalence of functors. Thus, hocolim_{O_{K}(G)} β_{K} ≃ hocolim_{O_{L}(G)} β_{L}

which proves (i).

Next we consider the case of subgroup sharp extensions. We use the same idea. Let L(H_{∗}(β_{K}; 𝔽_{p}) := lim_{\overrightarrow{O_{K}⟶G/L}} H_{∗}(β_{K}; 𝔽_{p}) be the left Kan extension of H_{∗}(β_{K}; 𝔽_{p})

for the inclusion O_{K}(G) ⟶ O_{L}(G) and denote by L^{i} the left derived functors of lim_{\overrightarrow{O_{K}⟶G/L}} H_{∗}(β_{K}; 𝔽_{p}). Since we can identify the under category O_{K} ⟶ G/L with

the orbit category O_{K_{L}}(L) and since , by assumption, K_{L} is subgroup sharp, L^{i} vanishes for i > 0 and L^{0} ≅ H_{∗}(β_{L}; 𝔽_{p}). An application of the composition of functor spectral sequence (cf. [6; Appendix II, 3.6]) proves the claim. □

4. Proofs of the statements.

In this section we want to prove our results stated in the introduction. We start with

Proof of Proposition 1.5. If C is a minimalization of P, then, by definition, C ⊂ C ∪ P is a sharp reduction. Moreover, since P_{H} is the collection of all nontrivial p-subgroups of H and since such collection are always subgroup sharp [4; 7.4], the claim follows from the propositions 3.2 and 3.3. □

In view of Proposition 1.5, the subgroup sharpness claimed in Theorem 1.3 and Theorem 1.4 is a consequence of the following two lemmas.

4.1 Lemma. Let G be a finite group.

(i) Let ρ : G ⟶ Gl(V) be a modular representation. Then I(ρ) is a minimalization of P. (ii) Let X be a G-space. Then, I(X) is a minimalization of P.

4.2 Lemma. Let G be a finite group such that p divides the order of G. Let C := Cen(E), Cen(A) denote the collection of centralizers of all nontrivial elementary abelian or of all nontrivial abelian subgroups of G. Then, C ⊂ C ∪ P^{c} is a strong reduction and P^{c} ⊂ C ∪ P^{c} is a subgroup sharp extension.

Proof of Lemma 4.1. For each subgroup K ⊂ G we denote by H_{K} := Iso_{G}(V^{K}) the isotropy group of V^{K}. Then, K ⊂ H_{K} and H_{K} ⊂ Iso_{G}(V^{H_{K}}) ⊂ Iso_{G}(V^{K}) = H_{K}. Hence, in the latter sequence all groups are equal. In particular, H_{(H_{K})} = H_{K}. Moreover, if P ∈ P, H ∈ I(ρ) and P ⊂ H, then Iso_{G}(H) ⊂ Iso_{G}(P) and H_{P} ⊂ H. This shows that I(ρ) ⊂ I(ρ) ∪ P is a minimalization. The proof of (ii) is analogously. □

Proof of Lemma 4.2. For a subgroup K ⊂ G we denote by E(K) the maximal elementary abelian subgroup of the center Z(K) of K. For P ∈ P we set H_{P} := C_{G}(E(P)) and for K ∈ Cen(E) \ P, we set H_{K} = K.

If H := C_{G}(E) ∈ Cen(E), then E ⊂ E(H) and H ⊂ C_{G}(E(H)) ⊂ C_{G}(E) = H. Hence, in the latter sequence, all groups are equal. If P ∈ P^{c} and H := C_{G}(E) ∈ Cen(E) such that P ⊂ H, then Z(P) ⊂ C_{G}(P) =: C is a p-Sylow subgroup and, in particular, E(H) ⊂ E(P). Thus H_{P} ⊂ C_{G}(E(H)) = H. This shows that Cen(E) ⊂ Cen(E) ∪ P^{c} is a strong reduction.

Moreover, if H := C_{G}(E) and P ⊂ H is p-centric in G or H, then E(H) ⊂ Z(P) and C_{H}(P) = C_{G}(P). That is to say that P is a p-centric subgroup of H if and only if it is a p-centric subgroup of G. Therefore, P_{H}^{c} is the collection of all p-centric subgroups of H which is subgroup sharp [4; 10.3]. This proves the second part of statement.

For Cen(A), the proof is analogously. We only have to replace E(H) by the center Z(H) of H. □

4.3 Remark.

(i) As the proofs and the Remark 3.4 shows, I := I(ρ), I(X) is M-subgroup sharp for a 𝔽_{p}[G]-module M if P as well as P_{H} for every H ∈ I are M-subgroup sharp. The arguments in [4; 7.4] show that this is true if every element of order p acts trivially on M. (ii) Again, the arguments in the proofs of both lemmas show that we can choose a smaller collection of isotropy subgroups of G. Let I_{P}(ρ) denote the collection of all isotropy subgroups of the subspaces V^{P} for P ∈ P. Then, I_{P}(ρ) is a minimalization of P and therefore M-subgroup sharp for M as above.

In the rest of this section we prove centralizer and normalizer sharpness properties for our collections.

4.4 Theorem. Let G be a finite group such that p divides the order of G.

(i) Let ρ : G ⟶ Gl(V) be a modular representation. Then, the collection I(ρ) is normalizer sharp. (ii) Let X be a G-space. Then, I(X) is normalizer sharp.

Proof. We use the techniques of [4; §7]. For a subgroup K ⊂ G we set H_{K} := Iso_{G}(V^{K}). For every nontrivial p-subgroup Q ⊂ G, the fixed-point set (X_{I(ρ}^{δ})^{Q} is the nerve of the category X_{I(ρ)^{Q}}^{δ} [4; 7.2]. And I(ρ)^{Q} is the full subcategory of all H ∈ I(ρ) such that Q ⊂ N_{G}(H). Obviously, the isotropy group H_{Q} is fixed, thus X_{I(ρ)^{Q}}^{δ} is nonempty. The natural transformations induced by the inclusions H ⊂ H_{H·Q} ⊃ H_{Q} show that the identity is homotopic to the constant functor. Therefore, X_{I(ρ)^{Q}}^{δ} is contractible.

Moreover, for each simplex σ := (H_{0} ⊂ H_{1} ⊂ ... ⊂ H_{n}) of X_{I(ρ)}^{δ}, the isotropy group G_{σ} = ⋂_{i} N_{G}(H_{i}) contains H_{0} and therefore an element of order p which acts trivially on 𝔽_{p}. This is sufficient to conclude that X_{I(ρ)}^{δ} is acyclic for the functors H_{∗}(G; −) [4; 6.10] and that I(ρ) is normalizer sharp. □

4.5 Remark. A long the same lines, one can proof that I(ρ) is subgroup sharp. We use the same notation as above. The fixed-point set (X_{I(ρ)}^{β})^{Q} is the nerve of the full subcategory of object (gH, G/H) such that Q ⊂ gHg^{−1}. Hence, it is nonempty, and, since there is a unique map (eH_{Q}, G/H_{Q}) ⟶ (gH, G/H) for each object, it has an initial element and is therefore contractible.

A typical simplex of X_{I(ρ)}^{β} has as its isotropy group an element of I(ρ) and contains therefore an element of order p. Analogously as above, this implies that I(ρ) is subgroup sharp.

4.6 Proposition. Let G be a finite group such that p divides the order of G.

(i) Let ρ : G ⟶ Gl(V) be a modular representation. Then, the collection P∪I(ρ) =: PI is centralizer sharp. (ii) Let X be G-space. Then, the collection P ∪ I(X) =: PI is centralizer sharp.

Proof. Again, H_{K} denotes Iso_{G}(V^{K}) respectively Iso_{G}(X^{K}). First we show that for every nontrivial p-subgroup Q the fixed-point sets (X̂_{PI}^{α})^{Q} are contractible. This fixed-point set is the nerve of the full subcategory whose objects are pairs (H, gC_{G}(H)) such that g^{−1}Qg ⊂ C_{G}(H). The fixed point set is nonempty since Q ⊂ C_{G}(Z(Q)). The zig-zag H ⊂ H_{H·Z(Q)} ⊃ H_{Z(Q)} establishes natural transformation between the associated functors. This shows that (X̂_{PF}^{α})^{Q} is contractible.

The same is true for (X̂_{P}^{α})^{Q} [4; 7.3]. Therefore, for a p-Sylow subgroup P ⊂ G, the inclusion j : X̂_{P}^{α} ⟶ X̂_{PI}^{α} is a weak P -equivalence and induces an isomorphism in P -equivariant Bredon homology for any coefficient functor Φ. Now we can apply a relative version of [4; 6.4] (see the lemma below) to show that j induces an isomorphism in G-equivariant Bredon homology for any coefficient functor for G. Since X̂_{P}^{α} is acyclic for H_{∗}(G; −) [4; 7.3], this is also true for X_{PI}^{α}. This shows that PI is centralizer sharp. □

4.7 Lemma. Let f : X ⟶ Y be a G-map between G-spaces. Let Φ be a coefficient functor for G and let K ⊂ G be a subgroup of index coprime to p. If f is an Φ|_{K}-equivalence as a K-map, then it is a Φ-equivalence as a G-map, too.

Proof. Let C denote the cone of f. Then

\begin{matrix}X & ⟶ & Y \\ ↓ & & ↓ \\ ∗ & ⟶ & C \end{matrix}

is a homotopy push out of G-spaces in the sense of [4; §4]; i.e. for every subgroup H ⊂ G the associated diagram of H-fixed subspaces is a homotopy pushout of spaces. The long exact sequence for P -equivariant Bredon homology of the above homotopy push out shows that C is acyclic as a K-space for Φ|_{K}. Hence, by [4; 6.4], the G-space C is acyclic for Φ and f as a G-map is a Φ-equivalence. Again this last step follows from the long exact sequence for G-equivariant Bredon homology. □

4.8 Corollary. Let G be a finite group such that p divides the order of G.

(i) Let ρ : G ⟶ Gl(V) be a modular representation. Then, the collection I(ρ) is centralizer sharp. (ii) Let X be G-space. Then, the collection I(X) is centralizer sharp.

Proof. This follows from Lemma 4.1 and its proof, Proposition 3.2 and Proposition 4.6. □

4.9 Remark. The arguments in the proof of Proposition 4.6 and Proposition 3.2 show that the centralizer decomposition for I(ρ) has the same centralizer sharpness properties as P. If M is 𝔽_{p}[G]-module, such that there exists an element 1 ≠ g ∈ G of order p which acts trivially on M, then P is M-centralizer sharp [4; 7.3] and so is I(ρ).

Finally we want to show that the collection Cen(E) and Cen(A) are normalizer sharp. This will finish the proofs of everything claimed in the introduction.

4.10 Theorem. The collections Cen(E) and Cen(A) are normalizer sharp.

Proof. We only prove this statement for C := Cen(E). A slight modification of the argument will work for Cen(A).

Let Q be a nontrivial p-subgroup. Then, (X_{C}^{δ})^{Q} is the nerve of the full subcategory of X_{C}^{δ} whose objects satisfy the condition Q ⊂ N_{G}(C_{G}(E). This category is nonempty since Q ⊂ C_{G}(E(Q). Moreover, Q acts on E′ := E(C_{G}(E)). The zig-zag

C_{G}(E) ⊂ C_{G}((E′)^{Q}) ⊃ C_{G}((E′)^{Q} · E(Q)) ⊂ C_{G}(E(Q))

shows that this fixed point set is contractible. Moreover, N_{G}(C_{G}(E)) contains an element of order p. Therefore, (X_{C}^{δ}) is acyclic for the coefficient functor H_{∗}(G; −)

[4] and C is normalizer sharp. □

4.11 Remark. Let M be a 𝔽_{p}[G]-module, such that there exists an element 1 ≠ g ∈ G of order p which acts trivially on M Then, for every E ∈ E, there exists an 1 ≠ h ∈ C_{G}(E) ⊂ N_{G}(C_{G}(E)) of order p which also acts trivially on M [4; 7.1]. This is sufficient to conclude that X_{Cen(E)}^{δ} is acyclic with respect to H_{∗}(G; M ⊗ −) [4; 6.10] and that Cen(E) is M-normalizer sharp.

5. Examples.

In this section we discuss some examples. Again, G denotes a finite group.

5.1 Polynomial invariants of finite groups. This example was used in [8] and [9] to construct spaces with polynomial mod-p cohomology and to prove uniqueness results for these spaces. Actually it was the motivating example for this note.

Let P′ denote the collection of all p-subgroups of G (including the trivial group). This collection is M-subgroup sharp for any 𝔽_{p}[G]-module M [4; 10.1 and the proof], even if we replace homology by cohomology. For cohomology, this is proved in [7].

Let ρ : G ⟶ Gl(V) be a modular representation, let I′ := I′(ρ) denote the collection of isotropy subgroups of all nontrivial subspaces of V (including V itself) and let I_{P′} := I_{P′}(ρ) denote the subcollection of all isotropy subgroups of the subspaces V^{P} for P ∈ P′. Then, I′ and I_{P′} are both M-subgroup sharp for every 𝔽_{p}[G]-module M (for homology as well as for cohomology).

Let ϕ : G ⟶ Gl(W) be another representation of G. The the polynomial algebra A := 𝔽_{p}[W] of formal polynomial function on W carries a G-action. For J = I′ or I_{P′}, the functor H_{A}^{∗} : O_{J}^{op}(G) ⟶ Ab defined by H_{A}^{∗}(G/H) := H^{∗}(H; A) (group cohomology) is isomorphic to H^{∗}(β_{J}; A) (twisted coefficients). We get the algebraic decomposition formula

lim_{\overleftarrow{J}}^{i}H_{A}^{∗} ≅ \{ \begin{matrix}H^{∗}(G; A) & for i = 0 \\ 0 & for i ≥ 1 \end{matrix}

And, specializing to H_{A}^{0}, this reads as

lim_{\overleftarrow{J}}^{i}𝔽_{p}[W]^{H} ≅ \{ \begin{matrix}𝔽_{p}[W]^{G} & for i = 0 \\ 0 & for i ≥ 1 \end{matrix}

5.2 Decompositions for the symmetric group Σ_{n}. The set X := {1, ..., n} given by the first n positive integers carries a Σ_{n}-action. For a p-subgroup P ⊂ Σ_{n} a P -orbit has length 1 or a power of p. Hence, the fixed-point set X^{P} is a subset of X of order r such that p divides n − r. And the isotropy subgroup of X^{P} is conjugate to Σ_{n−r}.

Since every subset of X of order n−r such that p|(n−r) appears as the fixed-point set of a p-subgroup, the collection C of all subgroups conjugate to some Σ_{m} ⊂ Σ_{n} where p|m is subgroup, centralizer and normalizer sharp.

Let Σ_{n} ⟶ Gl(n, 𝔽_{p}) denote the permutation representation. And let B := {e_{1}, ..., e_{n}} denote the standard basis of 𝔽_{p}^{n} =: V. For a subset B′ ⊂ B we denote by V (B′) ⊂ V the subspace generated by the sum of all elements in B′. The isotropy subgroup of V (B′) is conjugate to Σ_{r} × Σ_{n−r} where r is the order of B′. For a nontrivial p-subgroup P ⊂ Σ_{n} the fixed-point set V^{P} is the direct sum of subspaces V (B′) where B′ runs through the P -orbits of B. Since each of these orbits has length a power of p, the isotropy subgroup of V^{P} is conjugate to a subgroup of the form Σ_{p^{n_{1}}} ×...×Σ_{p^{n_{k}}} where p^{n_{1}} +...+p^{n_{k}} = n and where n_{1} ≥ 1. Let C denote the collection of all subgroups conjugate to a product of symmetric groups as described above. Each of these groups appears as the isotropy subgroup of the fixed-point set of some p-subgroup. Therefore, the collection C is subgroup, centralizer and normalizer sharp.

5.3 A decomposition for Gl(n, 𝔽_{p}). The general linear group G := Gl(n; 𝔽_{p}) acts on V := 𝔽_{p}^{n} by matrix multiplication. Again, we denote by B := {e_{1}, ..., e_{n}} the standard basis of V. Let V (r) denote the subspace generated by the first r basis vectors. The group G acts transitively on the set of r-dimensional subspaces. Therefore, the isotropy subgroup of a proper nontrivial subspace V′ ⊂ V is conjugate to the isotropy subgroup of V (r), where 1 ≤ r = dim_{𝔽_{p}}V′. And this is the subgroup G(r) of all matrices of the form

( \begin{matrix}1 & B \\ 0 & D \end{matrix} ), where D ∈ Gl(n − r, 𝔽_{p}) and where B is a r × (n − r) matrix over

𝔽_{p}. All these subgroups contain elements of order p. Therefore, the collection of all subgroups conjugate to some G(r), 0 < r < n is subgroup, centralizer and normalizer sharp.

5.4 Decompositions for finite groups associated to permutation representation. Let H ⊂ G be a subgroup of the finite group G. The G-action of G on the homogeneous space G/H induces an action of G on the vector space V := 𝔽_{p}[G/H] whose standard basis is given by the elements gH ∈ G/H. As in 5.2, we associate for each subset of B ⊂ G/H the 1-dimensional subspace V (B) ⊂ V generated by the sum of all elements in B. For a p-subgroup P ⊂ G the fixed-point set V^{P} is the direct sum of all subspaces V (B) where B runs through the P -orbits in G/H. And a subgroup K ⊂ G with P ⊂ K is contained in the isotropy subgroup of V^{P} if the obvious map P \G/H ⟶ K\G/H is a bijection. That is to say that, for all gH ∈ G/H and for all k ∈ K, kgH ⊂ P gH or , equivalently, such that k ∈ P · gHg^{−1}. The product is understood as the set of elements ph, p ∈ P and h ∈ gHg^{−1}, not as the subgroup generated by P and gHg^{−1}. The isotropy group of V^{P} is therefore the subgroup H_{P} := ⋂_{gH∈G/H} P · gHg^{−1}. The collection of all subgroups H_{P}, P ∈ P is subgroup, centralizer and normalizer sharp.

5.5 Decomposition associated to collections of subgroups. Let X := C be a collection of subgroups of G, such that there exists an element C ∈ C whose normalizer contains a p-Sylow subgroup of G. For a p-subgroup P ⊂ G, the isotropy subgroup of X^{P} is the intersection ⋂ N_{G}(C) taken over all subgroups C such that P ⊂ N_{G}(C). The collection of all subgroups of this form is nonempty and subgroup, centralizer and normalizer sharp.

If we choose C to be P itself then the collection of all intersection N_{G}(P_{1}) ∩ ... ∩ N_{G}(P_{k}) with nontrivial p-Sylow subgroup is subgroup, centralizer and normalizer sharp.

If we choose C to be the collection of all p-Sylow subgroups, then the collection of all intersections N_{G}(P_{1}) ∩ ... ∩ N_{G}(P_{k}) such that P_{1} ∩ ... ∩ P_{k} is a nontrivial intersection of p-Sylow subgroups is subgroup, centralizer and normalizer sharp.

If we choose X to be the set of all objects of the category X_{C}^{β} where C is again the collection of all p-Sylow subgroups, then the collection of all nontrivial intersections P_{1} ∩ ... ∩ P_{k}, P_{i} a p-Sylow subgroup for all i, is subgroup centralizer and normalizer sharp.

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Max-Planck-Institut für Mathematik, P.O.Box: 7280, D-53072 Bonn, Germany. e-mail : notbohm at mpim-bonn.mpg.de