HOMOLOGY DECOMPOSITIONS FOR

CLASSIFYING SPACES OF FINITE GROUPS

ASSOCIATED TO MODULAR REPRESENTATIONS

D. Notbohm

Abstract. For a prime p, a homology decomposition of 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 an 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. A list of examples 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 [9] and [10], 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 homology and sharp homology decompositions [3] [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. And this also works for homology with twisted coefficients.

Let M be an 𝔽_{p}[G]-module. If there exists an M-equivalence ϕ : hocolim_{D} F ⟶

BG, i.e. ϕ induces an isomorphism in H_{∗}(−; M)-homology (twisted coefficients), then this map is called a M-homology decomposition. We call this decomposition M-sharp, if

lim_{\overrightarrow{D}}_{i} H_{∗}(F; M) = \{ \begin{matrix}0 & for i ≥ 1 \\ H_{∗}(BG; M) & 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. Let M be an 𝔽_{p}[G]-module. We call a collection C subgroup M-sharp if the map b_{C} gives an M-sharp decomposition, centralizer M-sharp if a_{C} gives an M-sharp decomposition and normalizer M-sharp if d_{C} does so.

We will construct collections of subgroups associated to modular representations which are subgroup sharp. Let G be a finite group, V a finite dimensional 𝔽_{p}-vector space and ρ : G ⟶ Gl(V) a modular representation. Then we define I(ρ) to be the collection of all subgroups K of G with order divisible by p which appear as isotropy subgroups of nontrivial proper subspaces of V. And I′(ρ) is the collection of all subgroups with the same properties but which appear as isotropy groups of nontrivial subspaces. In particular, I′(V) contains the kernel of ρ.

1.2 Theorem. Let G be a finite group such that p divides the order of G. Let ρ : G ⟶ Gl(V) be a modular representation.

(i) The collection I′(ρ) is subgroup M-sharp for any 𝔽_{p}[G]-module M. (ii) If ρ is faithful, then I(ρ) is subgroup 𝔽_{p}-sharp.

For a G-space or a G-set X we define I′(X) and I(X) similarly as for a modular representation. I(X) is the collection of all subgroups of G with order divisible by p which appear as the isotropy group of nonempty proper subsets of X and I′(X) is the union of I(X) and the subgroup of all elements acting trivially on X. We say that G acts effectively, if this subgroup is the trivial group.

1.3 Theorem. Let G be a finite group such that p divides the order of G. Let P ⊂ G be a p-Sylow subgroup. Let X be a G-set such that the fixed point set X^{P} is nonempty.

(i) The collection I′(X) is subgroup M-sharp for any 𝔽_{p}[G]-module M. (ii) If G acts effectively on X, then I(X) is subgroup 𝔽_{p}-sharp.

For a collection C all the decompositions b_{C}, a_{C} and d_{C} are M-homology decompositions if one of them is one [3] (see Theorem 2.5). Thus, all the collection of the above theorems also establish centralizer and normalizer homology decompositions for suitable 𝔽_{p}[G]-modules. Actually, as shown in Section 4, our collections are normalizer M-sharp for suitable 𝔽_{p}[G]-modules M. As the Example 5.3 shows, we cannot expect to have centralizer sharpness, too.

If a collection C contains the group G itself, we do not get an interesting decomposition formula. 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. To avoid this, one should look for modular representations which are fixed-point free, i.e. V^{G} ≠ {0} respectively for fixed-point free G-sets.

If the representation ρ is faithful or if the G-action is effective, then the collections I′(ρ) and I′(X) contain the trivial group {e}. In this case the group G also appears in the subgroup decomposition diagram, namely as the automorphism group of the object G/{e}. In contrast, for the collections I(ρ) and I(X), all automorphism groups of objects of the associated orbit category are quotients of proper subgroups of G. And this is the reason why we also looked at I(ρ) and I(X) and why we think that they give more interesting decomposition formulas than the other two.

Both statements are an easy consequences of the following theorem which states subgroup sharpness for collections satisfying the first and one of the last two of the following conditions:

(P) Every nontrivial p-subgroup is contained in an element of C. (S) The collection is closed under taking intersections (wS) The intersection H := H_{1} ∩ H_{2} of two elements of C is contained in C if H is nontrivial with order divisible by p.

1.4 Theorem. Let G be a finite group whose order is divisible by p. Let C be a collection of subgroups of G.

(i) If C satisfies the conditions (P) and (S), then it is subgroup M-sharp for any 𝔽_{p}[G]-module M. (ii) If C satisfies the conditions (P) and (wS), it is subgroup 𝔽_{p}-sharp.

In Section 4, we also prove some normalizer sharpness properties for such collections.

Proof of Theorem 1.2 and Theorem 1.3. We only have to show that all the collections satisfy the correct conditions. For a p-subgroup Q ⊂ G, we have Q ⊂ Iso_{G}(V^{Q}). For a nontrivial subgroup Q ⊂ G, we have V^{Q} ≠ V if ρ is faithful and X^{Q} ≠ X if G acts effectively. This shows that all the collections satisfy condition (P).

For two subspaces V_{1}, V_{2} ⊂ V or two subset X_{1}, X_{2} ⊂ X, we have Iso_{G}(V_{1}+V_{2}) = Iso_{G}(V_{1}) ∩ Iso_{G}(V_{2}) and Iso_{G}(X_{1} ∪ X_{2}) = Iso_{G}(X_{1}) ∩ Iso_{G}(X_{2}). This shows that the collections I′(ρ) and I′(X) satisfy condition (S).

If ρ is faithful and {e} ≠ Q ⊂ Iso_{G}(V_{1}) ∩ Iso_{G}(V_{2}) := H, then V_{1} + V_{2} ≠ V, since the sum is fixed by a nontrivial subgroup, and H ∈ I(ρ). Hence, I(ρ) satisfies (wS). A similar argument works for an effective G-action. □

The method of proof for Theorem 1.4 can also be used to reconstruct the Jackowski-McClure decomposition via centralizers of nontrivial elementary abelian p-subgroups [8]. 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.

1.5 Theorem. The collections Cen(E) and Cen(A) are subgroup and normalizer 𝔽_{p}-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 full subcategory of the category of subgroups. In fact, it is the centralizer decomposition associated to the collection E. 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. That is, the collection E is centralizer sharp if and only Cen(E) is subgroup sharp. We intend to analyze this relation in more detail in a future paper.

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 and by P′ the collection of all 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. For I(ρ) and I(X) we replace P′ by P and for Cen(E) and Cen(A) by the collection P^{c} of all p-centric p-subgroups of G. A p-subgroup P ⊂ G is called p-centric if the center Z(P) ⊂ C_{G}(P) of P is a p-Sylow subgroup of the centralizer C_{G}(P).

The paper is organized as follows: In the next section we recall material 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 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.

We notice that the first part of Theorem 1.2 is already proved in [6] but by different methods.

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 an M-equivalence, M an 𝔽_{p}[G]-module, if it induces an isomorphism in H_{∗}(−; M)-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 Gset can be considered as a discrete G-space. Then, the homotopy colimit X := hocolim_{D} γ̃ is a G-space. It 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 an 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.

Let M be an 𝔽_{p}[G]-module. The E^{2}-term of the Bousfield-Kan spectral sequence for H_{∗}(−; M)-homology of hocolim_{D} γ can be identified with the Bredon homology

groups of hocolim_{D} γ̃ with coefficient functor H_{∗}(G; M ⊗−) [4; §3, 4.4 and 6.2]. And hocolim_{D} γ ⟶ BG is an M-sharp decomposition if and only if hocolim_{D} γ̃ is acyclic

with respect to H_{∗}(G; M ⊗ −).

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 an M-homology decomposition if b_{C} is an M-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 M-sharp if and only if (X_{C}^{β}) is acyclic with respect to H_{∗}(G; M ⊗ −).

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 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 an M-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 M-sharp if and only if X_{C}^{α} is acyclic with respect to H_{∗}(G; M ⊗ −).

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 an M-equivalence then we get an M-homology decomposition of BG. Let sdX_{C}^{δ} := hocolim_{\overline{s}S_{C}} δ̃_{C}. and let X_{C}^{δ} be 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 an M-sharp decomposition if and only if X_{C}^{δ} is acyclic with respect to H_{∗}(G; M ⊗ −).

By construction, K_{C} consists of the non degenerate simplices of X_{C}^{δ}, and both are G-equivariant homotopy equivalent.

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 M-ample if the G-equivariant map K_{C} ⟶ ∗ induces an M-equivalence (K_{C})_{hG} ⟶ BG.

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

3. Comparison of collections.

A collection C of subgroups of a finite group G is a partially ordered set and can be treated as a category denoted by S_{C}. This follows our convention to denote categories by boldface characters. Let S_{C} denote the nerve of BS_{C}. Then K_{C} ≃ S_{C}, since the simplices of K_{C} are just the non degenerate simplices of S_{C}.

An inclusion of collections H ⊂ L can be interpreted as a functor ι : S_{H} ⟶ S_{L}. For an element L ∈ L there is a standard over category ι→L and a standard under category L→ι, here denoted by S_{H}→L and L→S_{H}. Both of these categories come from posets. The first is the category associated to the set of elements of H containing L and the second is given by the poset H_{L} of all elements of H which are contained in L. The first is a set of subgroups of G, while H_{L} is actually a collection of subgroups of L. If all over categories or all under categories are contractible, Quillen’s Theorem A [11] implies that the functor ι : S_{H} ⟶ S_{L} induces equivalences S_{H} ≃ S_{L} and K_{H} ≃ K_{L}.

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. And, considered as a category, this is the under category S_{K}→H. Actually, we will compare H with H∪K and K with H∪K. In the latter case we like to 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. And let M be an 𝔽_{p}[G]-module.

(i) The collection H is called a sharp reduction of L if H ⊂ L and if for every L ∈ L \ H the over category L→S_{H} has an initial element. (ii) We say that L is an M-ample extension of K, if K ⊂ L and if K_{L} is an M-ample collection for all L ∈ L \ K. The collection is called a subgroup M-sharp extension if, in addition, K_{L} is a subgroup M-sharp collection for L.

3.2 Proposition. Let H ⊂ L be two collections of subgroups of a finite group G. Let M be an 𝔽_{p}[G]-module. If H is a sharp reduction of L, then the following holds:

(i) The map K_{H} ⟶ K_{L} is a G-equivariant homotopy equivalence and H is Mample if and only if L is M-ample. (ii) The maps X_{H}^{β} ⟶ X_{L}^{β} and X_{H}^{δ} ⟶ X_{L}^{δ} are G-equivariant homotopy equivalences.

In fact, Proposition 3.2 says that H has the same M-ampleness and the same subgroup and normalizer M-sharpness properties as L.

3.3 Proposition. Let K ⊂ L be two collections of subgroups of a finite group G. and let M be an 𝔽_{p}[G]-module.

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

Proof of Proposition 3.2. If H ⟶ L is a sharp reduction we can define a map F : L ⟶ L by F (L) := H_{L} where L ⟶ H_{L} is the initial element of L→S_{H}. This map is idempotent, respects inclusion, is G-equivariant and maps onto H. The Gequivariance follows, since gH_{L}g^{−1} = H_{gLg^{−1}}. Actually, H ⊂ L is a sharp reduction if and only if such a map exists.

The map allows us to construct G-equivariant left inverses s_{β} and s_{δ} for the inclusion functors i_{β} : X_{H}^{β} ⊂ X_{L}^{β} and i_{δ} : X^{δ} ⊂ X_{L}^{δ} and G-equivariant natural transformations id ⟶ i_{β}s_{β} and id ⟶ i_{δ}s_{δ}. As Lemma 3.4 below will show this is sufficient to prove part (ii).

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 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 first part follows from the second since, for any collection C, K_{C} and X_{C}^{δ} are G-equivariant homotopy equivalent (see 2.3). □

Since, in general, the centralizer C_{G}(L) is not contained in C_{G}(F (L)), we are not able to construct a G-equivariant left inverse for the functor i_{α} : X^{α} ⊂ X_{L}^{α}.

3.4 Lemma. Let i : C ⟶ D be a G-equivariant functor between small categories. which has a G-equivariant left inverse s : D ⟶ C. If there exists a G-equivariant natural transformation id ⟶ i ∘ s, 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 (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 non equivariant 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 M-ample extension. We want to show that K_{K} is M-ample if and only K_{L} is or, equivalently, that b_{K} : hocolim_{O_{K}(G)} β_{K} ⟶ BG

is an M-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]).

In particular, this equivalence is an M-equivalence for any 𝔽_{p}[G]-module M.

Since all objects as well as all morphisms in the under category O_{K} ⟶ G/L are given by G–maps, there exists a natural transformation T : L(β_{K}) ⟶ β_{L}. 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. Since the collection K_{L} is M-ample the natural transformation T consists of M-equivalences. This establishes an isomorphism between the E_{2}- pages of the Bousfield-Kan spectral sequences for H_{∗}(−; M)-homology. Hence, hocolim_{O_{L}(G)} L(β_{K}) ⟶ hocolim_{O_{L}(G)} β_{L} is an M-equivalence as well as hocolim_{O_{K}(G)} β_{K} ⟶

hocolim_{O_{L}(G)} β_{L}. This proves part (i).

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

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}; M). 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 M-sharp, L^{i} vanishes for i > 0 and L^{0} ≅ H_{∗}(β_{L}; M). An application of the composition of functor spectral sequence (cf. [7; Appendix II, 3.6]) proves the claim. □

4. Proof of Theorem 1.4 and Theorem 1.5.

The proof of the subgroup sharpness in both theorems is a consequence of the following two lemmas. For two collections H and CK we denote by HK := H ∪ K the union of both.

4.1 Lemma. Let G be a finite group and let M be an 𝔽_{p}[G]-module. Let C be a collection of subgroups.

(i) If C satisfies the conditions (P) and (S), then C ⊂ CP′ is a sharp reduction and P′ ⊂ CP′ is a subgroup M-sharp extension. (ii) If C satisfies the conditions (P) and (wS), then C ⊂ CP is a sharp reduction and P ⊂ CP is a subgroup 𝔽_{p}-sharp extension.

4.2 Lemma. Let G be a finite group such that p divides the order of G. For C = Cen(E) or Cen(A), the inclusion C ⊂ C ∪ P^{c} is a sharp reduction and P^{c} ⊂ C ∪ P^{c} is a subgroup 𝔽_{p}-sharp extension.

Proof of Lemma 4.1. For each (nontrivial) p-subgroup Q ⊂ G we denote by H_{Q} the intersection of all elements H ∈ C which contain Q. Because of condition (P), the subgroup H_{Q} is nonempty, and, because of condition (S) respectively (wS), H_{Q} ∈ C. Moreover, Q ⊂ H_{Q} is an initial element of Q→SC, which shows that C ⊂ CP′ respectively C ⊂ CP is a sharp reduction.

For any H ∈ C, the collection P′_{H} is the collection of all p-subgroups and P_{H} the collection of all nontrivial p-subgroups of H. The collection P′_{H} is subgroup M-sharp for any 𝔽_{p}[G]-module M and the P_{H} is subgroup 𝔽_{p}-sharp [4]. Hence, the inclusion P ⊂ CP′ is a subgroup M-sharp extension and the inclusion P ⊂ CP a subgroup 𝔽_{p}-sharp extension. □

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 P ⊂ H_{P} is an initial element of P →S_{Cen(E)} and that Cen(E) ⊂ Cen(E)P^{c} is a sharp 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. Hence, P_{H}^{c} is the collection of all p-centric subgroups of H and is therefore subgroup 𝔽_{p}-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. Let ρ : G ⟶ Gl(V) be a modular representation. Let I_{P}(ρ) (I_{P′}) denote the collection of all isotropy groups of the fixed-point set V^{Q} where Q ∈ P. (Q ∈ CP′). These are collection which in general are smaller than I(ρ) respectively I′(ρ). For any Q ∈ P′ the over categories Q→S_{I_{P′} (ρ)} have an initial element given by Q ⟶ Iso_{G}(V^{Q}). Therefore, I_{P}(ρ) ⟶ I_{P}(ρ) ∪ P is a sharp reduction. If ρ is faithful, the same holds I_{P}(ρ) ⟶ I_{P}(ρ) ∪ P. And as the proofs show, Theorem 1.2 is also true for these collections.

Obviously, the analogue results for an (effective) G-action on a space X are also true, if X^{P} ≠ ∅ for a p-Sylow subgroup P ⊂ G.

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

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

(i) If C satisfies the conditions (P) and (S) then C is normalizer M-sharp for any 𝔽_{p}[G]-module M. (ii) If C satisfies (P) and (wS), then C is normalizer 𝔽_{p}-sharp.

4.5 Corollary. Let G be a finite group with order divisible by p. Let ρ : G ⟶ Gl(V) be a modular representation. Then, the collection I′(ρ) is normalizer M-sharp for any 𝔽_{p}[G]-module. And, if ρ is faithful, I(ρ) is normalizer 𝔽_{p}-sharp.

The analogue results holds obviously for a G-action on a space X if X^{P} ≠ ∅ for a p-Sylow subgroup P ⊂ G.

Proof of Theorem 4.4. We use the techniques of [4; §7]. We first assume that C satisfies (wS). For a nontrivial p-subgroup Q ⊂ G we again denote by H_{Q} the intersection of all elements of C which contain Q.

For any nontrivial p-subgroup Q ⊂ G, the fixed-point set (X_{C}^{δ})^{Q} is the nerve of the category (X_{C}^{δ})^{Q} [4; 7.2]. And C^{Q} is the full subcategory of all H ∈ C such that Q ⊂ N_{G}(H). Obviously, the group H_{Q} is fixed, thus X_{I(ρ)^{Q}}^{δ} is nonempty. The natural transformations induced by the inclusions K ⊂ H_{K·Q} ⊃ H_{Q} show that the identity is homotopic to the constant functor. Therefore, (X_{C}^{δ})^{Q} is contractible.

Moreover, for each simplex σ := (H_{0} ⊂ H_{1} ⊂ ... ⊂ H_{n}) of X_{C}^{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_{C}^{δ} is acyclic for the functors H_{∗}(G; −) [4; 6.10] and that C is normalizer 𝔽_{p}-sharp which proves part (ii).

If C satisfies (S), then the category X_{C}^{δ} has an initial element, namely the isotropy group of V and X_{C}^{δ} is contractible. In particular, this and the above argument show that X_{C}^{δ} is acyclic for the functors H_{∗}(P; M ⊗ −), where P ⊂ G is a p-Sylow subgroup [4; 4.8]. By [4; 6.4], this implies that X_{C}^{δ} is acyclic for H_{∗}(G; M ⊗ −) and that C is normalizer M-sharp. □

4.6 Remark. A long the same lines, one can proof that C is subgroup 𝔽_{p}-sharp. We use the same notation as above. The fixed-point set (X_{C}^{β})^{Q} is the nerve of the full subcategory of all objects (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_{C}^{β} has as its isotropy group an element of C and contains therefore an element of order p. Analogously as above, this implies that I(ρ) is subgroup sharp.

For the collection I_{P}(ρ) (see Remark 4.3) we can improve Corollary 4.5.

4.7 Theorem. Let G be a finite group such that p divides the order of G, let ρ : G ⟶ Gl(V) be a modular representation and let M be an 𝔽_{p}[G]-module such that there exists a nontrivial element of order p in G acting trivially on M. Then, the collection I_{P}(ρ) is normalizer M-sharp.

Proof. For any nontrivial p-subgroup Q ⊂ G, we set H_{Q} := Iso_{G}(V^{Q}). Analogously as above follows that (X_{C}^{δ})^{Q} is contractible. For each simplex σ := (H_{0} ⊂ H_{1} ⊂ ... ⊂ H_{n}) of X_{C}^{δ}, let Q_{i} ⊂ H_{i} be a p-Sylow subgroup, such that Q_{i} ⊂ Q_{i+1}. Then, H_{i} = Iso_{G}(V^{Q_{i}}), and the centralizer C_{G}(Q_{n}) centralizes all Q_{i}’s, normalizes all H_{i}’s and is contained in the isotropy group G_{σ} = ⋂_{i} N_{G}(H_{i}) of σ. By [4; 7.1] there a nontrivial element of order p in G_{σ} acting trivially on M. And by [4; 6.10], this implies that I_{P}(ρ) is normalizer M-sharp. □

Finally we want to show that the collection Cen(E) and Cen(A) are normalizer sharp.

4.8 Theorem. Let G be a finite group whose order is divisible by p. Let M be an 𝔽_{p}[G]-module such that there exists a nontrivial element of order p acting trivially on M. Then, the collections Cen(E) and Cen(A) are normalizer M-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, by [4; 7.1], N_{G}(C_{G}(E)), actually C_{G}(E), contains an element of order p acting trivially on M. Therefore, (X_{C}^{δ}) is acyclic for the coefficient functor H_{∗}(G; M ⊗−) [4; 6.10] and C is normalizer M-sharp. □

5. Examples.

In this section we discuss some examples. We do not claim originality for the decomposition described in the following examples. Some of them might already be in the literature or known to experts.

5.1 Polynomial invariants of finite groups. This example was used in [9] and [10] 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. It is also discussed in [6].

For a finite group G, the collection P′ is subgroup M-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 [8].

Let ρ : G ⟶ Gl(V) be a modular representation. The collections I′(ρ) and I_{P′}(ρ) are both subgroup M-sharp for every 𝔽_{p}[G]-module M. For homology this follows from Theorem 1.2, and for cohomology one has to dualize all the arguments in the proof of Theorem 1.2.

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′(ρ) or I_{P′}(ρ), 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 r such that p|(n − r) appears as the fixed-point set of a p-subgroup, the collection of all subgroups conjugate to some Σ_{m} ⊂ Σ_{n}, where p|m, is subgroup and normalizer 𝔽_{p}-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 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 C of all subgroups conjugate to some G(r), 0 < r < n is subgroup and normalizer sharp.

The centralizer of G(r) in G is the center of G which is a cyclic group of order p − 1. Therefore, the values of the functor α_{C} are mod-p acyclic and the collection C is not centralizer 𝔽_{p}-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 and normalizer 𝔽_{p}-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 nontrivial 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 nontrivial subgroups of this form is nonempty and subgroup and normalizer 𝔽_{p}-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 and normalizer 𝔽_{p}-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 and normalizer 𝔽_{p}-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 and normalizer 𝔽_{p}-sharp.

References.

[1] D. Benson, Representations and cohomology II : Cohomology of groups and modules, Cambridge University Press, Cambridge, New York, Melbourne (1991).

[2] A. Bousfield & D. Kan, Homotopy limits, completion and localizations, SLNM 304, Springer Verlag, 1972.

[3] W.G. Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997), 783-804.

[4] W.G. Dwyer, Sharp homology decompositions for classifying spaces of finite groups, in: Group representations : Cohomology, group actions and topology, Ed.: A. Adem et al. Seattle (1996), 197-220.

[5] W.G. Dwyer & D. Kan, A classification theorem for diagrams of simplicial sets, Topology 23 (1984), 139-154.

[6] W.G. Dwyer & C.W. Wilkerson, A new finite loop space at the prime two, J. AMS 6 (1993), 37-63.

[7] P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Springer Verlag.

[8] S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992), 113-132.

[9] D. Notbohm, Topological realzation of a family of pseudo reflection groups, Fundam. Math. 155 (1998), 1-31.

[10] D. Notbohm, Spaces with polynomial cohomology, Math. Proc. Camb. Phil. Soc. 9 (1999), 55-99.

[11] D. Quillen, Higher algebraic K-theory I,, in: Algebraic K-Theory I, Lect. Notes in Math. 341, Springer, Berlin (1973), 85-147.

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