Extensions of G-posets and Quillen’s complex

Yoav Segev and Peter Webb*

ABSTRACT We develop techniques to compute the homology of Quillen’s com-

plex of elementary abelian p-subgroups of a finite group in the case where the group has a normal subgroup of order divisible by p. The main result is a long exact sequence relating the homologies of these complexes for the whole group, the normal subgroup, and certain centralizer subgroups. The proof takes place at the level of partially-ordered sets. Notions of suspension and wedge product are considered in this context, which are analogous to the corresponding notions for topological spaces. We conclude with a formula for the generalized Steinberg module of a group with a normal subgroup, and give some examples.

Subject Classification: Primary 20D30; Secondary 05E25, 06A09, 20C20, 51E25.

0. Introduction

Let G be a finite group and p a prime. Let A_{p}(G) be the Quillen complex of G at the prime p. A_{p}(G) is the order complex of the poset (= partially ordered set) of all non-trivial elementary abelian p-subgroups of G (see §1).

Let N be a normal subgroup of G. We denote by A_{p}(G)_{N} the poset obtained by adding to A_{p}(G) an additional element, say 0, so that 0 < A for all A ∈ A_{p}(G) with A ∩ N ≠ 1. Our main result is the following.

MAIN THEOREM. Let G be a finite group and p a prime. Suppose that N is a normal subgroup of G such that p divides |N|. Further except in (1) assume that if A is an elementary abelian p-subgroup of G with A ∩ N = 1 then A is cyclic, and let M = {A ∈ A_{p}(G) | A ∩ N = 1}.

(1) There exists a long exact sequence of ℤG-modules

⋯ → H̃_{n}(A_{p}(N))⟶^{ι_{∗}}H̃_{n}(A_{p}(G))⟶^{κ_{∗}}H̃_{n}(A_{p}(G)_{N})⟶^{s}H̃_{n−1}(A_{p}(N)) → ⋯

where ι_{∗} and κ_{∗} are induced by the obvious inclusion maps ι and κ.

* Partially supported by the NSF.

(2) There is a G-homotopy equivalence

A_{p}(G)_{N} ≃_{G} ⋁_{A∈M} ΣA_{p}(C_{N}(A))

where ⋁ denotes a wedge product and Σ denotes suspension.

(3) For n ≥ 0

H̃_{n}(A_{p}(G)_{N}) ≅ ⨁_{A∈M} H̃_{n−1}(A_{p}(C_{N}(A)))

≅ ⨁_{A∈M up to conjugacy} H̃_{n−1}(A_{p}(C_{N}(A))) ↑_{N_{G}(A)}^{G}

as ℤG-modules.

(4) With respect to the first isomorphism of (3) the map s of (1) is given by

s = ⨁_{A∈M}(ι_{A})_{∗}

where (ι_{A})_{∗} is the map on homology induced by the inclusion ι_{A} : A_{p}(C_{N}(A)) → A_{p}(N).

We will prove the Main Theorem in the abstract setting of a poset P having a subposet Q satisfying the two conditions that the elements of Q form an ideal in the opposite poset P^{op}, and for each p ∈ P there exists q ∈ Q with q ≥ p. In this situation we will call P an extension of Q, and in the context of the Main Theorem we will take P = A_{p}(G) and Q = {A ∈ A_{p}(G) | A ∩ N ≠ 1}. We give our sharpest results when P − Q consists of minimal elements of P, and in this generality Theorem 2.5 gives an inductive set-up for dealing with the homology of a poset.

After proving the Main Theorem in section 3, we consider the generalized Steinberg module St_{p}(G) in the situation that G has a normal subgroup N of order divisible by p. Because the computation of St_{p}(G) does not require such detailed information as the homology of A_{p}(G), we are able to remove the condition that A ∈ A_{p}(G), A ∩ N = 1 implies A is cyclic which was present in part of the Main Theorem, and obtain a result in generality. This result is Theorem 4.2. We conclude in section 5 with two worked examples.

1. The Mayer-Vietoris sequence for an extension of posets

We first describe a poset construction which gives rise to a Mayer-Vietoris sequence. Let P be a poset. The order complex of P is the simplicial complex ΔP whose simplices are the finite chains of elements of P. We may occasionally omit the symbol Δ, especially when considering the homology H_{n}(P), by which we mean H_{n}(ΔP). Given a subposet Q ≤ P and p ∈ P we denote by Q_{≥p} the subposet of P with elements {q ∈ Q | q ≥ p}. We define Q_{>p}, Q_{<p} etc. in a similar way. An ideal of P is a subposet ∅ ≠ I ≤ P such that if i ∈ I and p ≤ i then p ∈ I.

Let Q ≤ P be a subposet. We say that P is an extension of Q if Q is an ideal of the opposite poset P^{op}, and for all p ∈ P, Q_{≥p} ≠ ∅. We will further say that P is an extension of Q by minimal elements if P is an extension of Q and for all p ∈ P, either p ∈ Q or p is a minimal element of P. If P is an extension of Q we denote by P_{Q} the following poset. The elements of P_{Q} are P together with one additional element 0_{Q}. The order relation in P_{Q} is the following. Given x, y ∈ P_{Q}, x < y if and only if either x, y ∈ P and x < y in P, or x = 0_{Q} and y ∈ Q. We denote by Q_{Q} ≤ P_{Q} the similarly constructed poset with elements Q ∪ {0_{Q}}. If P is a G-poset and Q is G-invariant, then P_{Q} becomes a G-poset by letting G fix 0_{Q}.

Turning to our conventions for homology, we first mention that unless otherwise specified all homology groups are taken with coefficients in ℤ. Let K be a simplicial complex. We denote by C_{n}(K) the simplicial chain group of K at dimension n (with coefficients in ℤ), n ≥ 0. C̃_{n}(K) denotes the augmented simplicial chain group of K at dimension n, n ≥ −1. So C̃_{n}(K) = C_{n}(K) for all n ≥ 0, and C̃_{−1}(K) = ℤ. By Z_{n}(K) resp. B_{n}(K) resp. H_{n}(K) we denote the group of n-cycles resp. n-boundaries resp. n-homology group. The notation Z_{∗}(K), H_{∗}(K), Z̃_{n}(K), H̃_{n}(K), H̃_{∗}(K) etc. is now clear. Given a cycle z ∈ Z̃_{n}(K) we denote by [z] = z + B̃_{n}(K) the corresponding element in H̃_{n}(K).

(1.1) PROPOSITION. Let P be an extension of Q. Then

(1) ΔP_{Q} = ΔP ∪ ΔQ_{Q} and ΔP ∩ ΔQ_{Q} = ΔQ.

(2) There is a long exact Mayer-Vietoris reduced homology sequence

⋯ → H̃_{n}(Q)⟶^{ι_{∗}}H̃_{n}(P)⟶^{κ_{∗}}H̃_{n}(P_{Q})⟶^{r}H̃_{n−1}(Q) → ⋯

where ι_{∗}, κ_{∗} are the maps on homology induced by the obvious inclusion maps ι, κ and r is given as follows. If α ∈ C̃_{n}(P) and β ∈ C̃_{n}(Q_{Q}) are such that ∂(α + β) = 0, then r([α + β]) = [∂α], where ∂ is the differential map of P_{Q}. In case P is a G-poset and Q is G-invariant, the Mayer-Vietoris sequence is a sequence of ℤG-modules.

Proof. The Mayer-Vietoris sequence needs no proof. We only mention that as Q_{Q} is contractible, H̃_{∗}(Q_{Q}) = 0.

2. The structure of P_{Q}

The use of the Mayer-Vietoris sequence just described is considerably enhanced by the fact that we are able to give an explicit description of the space ΔP_{Q} in case P is an extension of Q by minimal elements, showing that it is homotopic to a wedge of suspensions of certain other posets. To define what we mean by this, we make definitions at the level of posets which copy well-known topological constructions.

First let R be a poset. In this paper by the suspension of R we mean a poset

ΣR = R ∪ {0_{R}, 0′_{R}}

where 0_{R} and 0′_{R} are two new symbols, and where the order relation is as follows. For x, y ∈ ΣR we define x ≤ y if and only if either x = 0_{R} and y ≠ 0′_{R}, or x = 0′_{R} and y ≠ 0_{R}, or x, y ∈ R and x ≤ y in R. In case R is a G-poset then G acts on ΣR by fixing 0_{R} and 0′_{R}.

To describe also the action of G on the homology of R and of ΣR we introduce the following notation. Given an (n − 1)-simplex s = (r_{0} < r_{1} < ⋯ < r_{n−1}) of R and r ∈ R with r < r_{0}, we denote r ∗ s = (r < r_{0} < r_{1} < ⋯ < r_{n−1}). Let n ≥ 1. Given a cycle z ∈ Z̃_{n−1}(R), write z = ∑_{i=1}^{m} n_{i}s_{i}, with s_{i} an (n − 1)-simplex of R. We write 0_{R} ∗ z = ∑_{i=1}^{m} n_{i}(0_{R} ∗ s_{i}) ∈ C̃_{n}(Σ(R)) and similarly we define 0′_{R} ∗ z ∈ C̃_{n}(Σ(R)). We write Σ(z) = 0_{R} ∗ z − 0′_{R} ∗ z.

(2.1) PROPOSITION. Suppose that R is a G-poset.

(1) There is a G-equivariant homeomorphism Δ(ΣR) ≅_{G} ΣΔ(R).

(2) For n ≥ 1, if z ∈ Z̃_{n−1}(R) then ∂(0_{R} ∗ z) = ∂(0′_{R} ∗ z) = z, and hence Σ(z) ∈ Z̃_{n}(ΣR). Here ∂ is the differential map of Σ(R).

(3) The map H̃_{n−1}(R) → H̃_{n}(ΣR) given by [z] → [Σ(z)] is an isomorphism of ℤGmodules.

Proof. In part (1) the action of G on ΣΔ(R) fixes the suspension coordinate and acts on Δ(R) in the given way. The homeomorphism is immediate, since each of the subposets R ∪ {0_{R}} and R ∪ {0′_{R}} is a cone on R with vertex fixed by G. The description in parts (2) and (3) is well-known.

Continuing in this vein, let {R_{t} | t ∈ T } be a set of posets indexed by some set T . By a wedge of suspensions of the R_{t} we mean a poset

X = ⋃_{t∈T} (R_{t} × {t}) ∪ T ∪ {0}

which we will denote by ⋁_{t∈T} ΣR_{t}. We define a partial order on this set as follows. For t ∈ T define

j_{t} : R_{t} × {t} → R_{t}

by j_{t}(r, t) = r. If x, y ∈ X we put x < y if and only if one of the following holds:

(i) there exists t ∈ T such that x, y ∈ R_{t} × {t} and j_{t}(x) < j_{t}(y), (ii) x = t ∈ T and y ∈ R_{t} × {t}, (iii) x = 0 and y ∉ T ∪ {0}.

The point about using the set R_{t} × {t} in the above construction is that it guarantees that all these sets are disjoint as t ranges through T . However, when no confusion may arise we identify R_{t} with R_{t} × {t} via j_{t}. We do this in the next result.

(2.2) PROPOSITION. Let {R_{t} | t ∈ T } be posets. Then

(1) Δ(⋁_{t∈T} ΣR_{t}) ≅ ⋁_{t∈T} ΣΔ(R_{t}), and

(2) for n ≥ 1 the map

μ : ⨁_{t∈T} H̃_{n−1}(R_{t}) → H̃_{n}( ⋁_{t∈T} ΣR_{t})

defined by μ(∑_{t∈T} [z_{t}]) = ∑_{t∈T} [t ∗ z_{t} − 0 ∗ z_{t}] is an isomorphism, where for all t ∈ T , z_{t} ∈ Z̃_{n−1}(R_{t}).

Proof. (1) is immediate from the definitions and (2) is well-known.

We will apply this construction in the situation where P is an extension of Q by minimal elements. We take the indexing set T to be the set M = P − Q, which consists of minimal elements of P, and the posets R_{t} are the P_{>m}, m ∈ M. In this situation, if P is a G-poset and Q is G-invariant we may define a group action on the poset ⋁_{m∈M} ΣP_{>m} by

\begin{aligned}g(p, m) & = (gp, gm),  for (p, m) ∈ P_{>m} × \{m\}, \\ gm & = gm,  m ∈ M, \\ g0 & = 0, \end{aligned}

and now the homology groups of this poset become ℤG-modules. There is also a group action on the simplicial complex

⋁_{m∈M} ΣΔ(P_{>m})

in which if x ∈ ΣΔ(P_{>m}) and g ∈ G then gx ∈ ΣΔ(P_{>gm}).

(2.3) PROPOSITION. Suppose that P is a G-poset which is an extension of Q by minimal elements where Q is G-invariant. Put M = P − Q.

(1) There is a G-equivariant homeomorphism

Δ( ⋁_{m∈M} ΣP_{>m}) ≅_{G} ⋁_{m∈M} ΣΔ(P_{>m}).

(2) For n ≥ 1 the group ⨁_{m∈M} H̃_{n−1}(P_{>m}) acquires the structure of an induced ℤG-

module

⨁_{m∈M} H̃_{n−1}(P_{>m}) ≅ ⨁_{m∈[G\M]} H̃_{n−1}(P_{>m}) ↑_{G_{m}}^{G}

where G_{m} is the stabilizer of m in G. The mapping

μ : ⨁_{m∈M} H̃_{n−1}(P_{>m}) → H̃_{n}( ⋁_{m∈M} ΣP_{>m})

of 2.2 is an isomorphism of ℤG-modules.

Proof. (1) is simply 2.2(1), with the observation that the homeomorphism is Gequivariant.

In (2) the ℤG-module structure of ⨁_{m∈M} H̃_{n−1}(P_{>m}) comes from the ℤG-module structure of ⨁_{m∈M} C̃_{n−1}(P_{>m}) in which g(p_{0} < p_{1} < ⋯ < p_{n−1}) = (gp_{0} < gp_{1} < ⋯ < gp_{n−1}) where (p_{0} < p_{1} < ⋯ < p_{n−1}) is an (n − 1)-simplex of P_{>m} and (gp_{0} < gp_{1} < ⋯ < gp_{n−1}) is an (n − 1)-simplex of P_{>gm}. Thus ⨁_{m∈M} H̃_{n−1}(P_{>m}) is the direct sum of subspaces permuted by G in which the stabilizer of each subspace is G_{m}, and the orbits of G on these subspaces are in bijection with G\M. Thus with this action ⨁_{m∈M} H̃_{n−1}(P_{>m}) is an induced module as claimed. It is evident from the definition that μ is a homomorphism of ℤG-modules as claimed.

We now show the relevance of the wedge of suspensions construction in the situation where P is an extension of Q by minimal elements. As before we put M = P − Q. Define

j : ⋁_{m∈M} ΣP_{>m} → P_{Q}

as follows. Firstly j(0) = 0_{Q}, secondly j(m) = m for all m ∈ M, and thirdly, for x ∈ P_{>m} × {m}, we define j(x) = j_{m}(x), where j_{m} was defined before 2.2.

(2.4) THEOREM. Suppose that P is a G-poset which is an extension of Q by minimal elements where Q is G-invariant. Put M = P − Q.

(1) j : ⋁_{m∈M} ΣP_{>m} → P_{Q} is a G-homotopy equivalence.

(2) For n ≥ 1 the map

μ : ⨁_{m∈M} H̃_{n−1}(P_{>m}) → H̃_{n}(P_{Q})

is a ℤG-module isomorphism, where

μ( ∑_{m∈M}[z_{m}]) = ∑_{m∈M}[m ∗ z_{m} − 0_{Q} ∗ z_{m}].

Here z_{m} ∈ Z̃_{n−1}(P_{>m}) for all m ∈ M.

Proof. First note that j is a G-equivariant map of posets. We show that for each p ∈ P_{Q}, j^{−1}((P_{Q})_{≤p}) is G_{p}-contractible. Then (1) follows from Theorem 1 in [6].

Case 1. There exists no m ∈ M with m ≤ p. In this case j^{−1}((P_{Q})_{≤p}) = {0} is evidently G_{p}-contractible.

Case 2. p ∈ M. In this case j^{−1}((P_{Q})_{≤p}) = {p} is also G_{p}-contractible.

Case 3. p ∉ M and there exists m ∈ M such that m < p. Let M_{p} = {m ∈ M | m < p} and X = ⋁_{m∈M} ΣP_{>m}. Notice now that j^{−1}((P_{Q})_{≤p}) = ⋃_{m∈M_{p}} X_{≤(p,m)} is a union of posets, each of which has a unique maximal member (p, m) and which pairwise intersect in {0}. Such a poset is evidently contractible. For convenience, let Y = j^{−1}((P_{Q})_{≤p}) and Z = {(p, m) | m ∈ M_{p}} ∪ {0} ⊆ X. Then the two maps

\begin{matrix}ψ : & Y → Z \\ ϕ : & Z → \{0\} \end{matrix}

defined by ψ(0) = 0 and ψ(y) = (p, m) if 0 ≠ y ∈ X_{≤(p,m)} and ϕ(p, m) = ϕ(0) = 0 are G_{p}-equivariant and satisfy y ≤ ψ(y), z ≥ ϕ(z) always. Therefore by [6, 1.2] ψ and ϕ are G_{p}-homotopy equivalences which show that Y is G_{p}-contractible.

Part (2) follows from (1) and 2.3.

We summarize the situation so far by combining Proposition 1.1 with Theorem 2.4 and a little bit more.

(2.5) THEOREM. Suppose that P is a G-poset which is an extension of Q where Q is G-invariant. Put M = P − Q, and in (2) – (4) assume that M consists of minimal elements.

(1) There is a long exact sequence of ℤG-modules

⋯ → H̃_{n}(Q)⟶^{ι_{∗}}H̃_{n}(P)⟶^{κ_{∗}}H̃_{n}(P_{Q})⟶^{r}H̃_{n−1}(Q) → ⋯

where ι_{∗}, κ_{∗} are the maps on homology induced by the obvious inclusion maps ι, κ.

(2) P_{Q} ≃_{G} ⋁_{m∈M} ΣP_{>m}.

(3) For all n ≥ 0

\begin{aligned}H̃_{n}(P_{Q}) & ≅ ⨁_{m∈M} H̃_{n−1}(P_{>m}) \\ & ≅ ⨁_{m∈[G\M]} H̃_{n−1}(P_{>m}) ↑_{G_{m}}^{G} \end{aligned}

as ℤG-modules.

(4) With respect to the first isomorphism of (3) the map r of (1) is given by

⨁_{m∈M}(ι_{m})_{∗}

where ι_{m} : P_{>m} → Q is the inclusion map.

Proof. Part (1) is 1.1 and parts (2) and (3) come from 2.3 and 2.4. We now prove part (4). We must show that in the long exact sequence

⋯ → H̃_{n}(Q)⟶^{ι_{∗}}H̃_{n}(P) ⟶^{μ^{−1}κ_{∗}} ⨁_{m∈M} H̃_{n−1}(P_{>m})⟶^{rμ}H̃_{n−1}(Q) → ⋯

the map rμ is given by

rμ = ⨁_{m∈M}(ι_{m})_{∗}

where μ is the map of 2.4(2).

Let h = ∑_{m∈M}[z_{m}] ∈ ⨁_{m∈M} H̃_{n−1}(P_{>m}). Then by 2.4(2), μ(h) = ∑_{m∈M}[m ∗ z_{m} − 0_{Q} ∗ z_{m}]. Notice now that m ∗ z_{m} ∈ P and 0_{q} ∗ z_{m} ∈ Q_{Q}. Hence by 1.1(2) and 2.1(2), rμ(h) = ∑_{m∈M} r([m ∗ z_{m} − 0_{Q} ∗ z_{m}]) = ∑_{m∈M}[∂(m ∗ z_{m})] = ∑_{m∈M}[z_{m}].

We conclude this section with a discussion of the multiple connectivity of the complexes P and P_{Q}. For each simplicial complex Δ we denote by Δ^{(n)} the n-skeleton of Δ.

(2.6) LEMMA. Let P be an extension of Q. If Q is n-connected then the inclusion map ι : ΔP^{(n)} → ΔP_{Q}^{(n)} is a homotopy equivalence, and P is n-connected if and only if P_{Q} is n-connected.

Proof. For p ∈ P, ι^{−1}((P_{Q})_{≥p}) = P_{≥p} is contractible, while ι^{−1}((P_{Q})_{≥0_{Q}}) = Q is n-connected. Hence Lemma 4.3 in [2] completes the proof.

(2.7) COROLLARY. Let P be an extension of Q by minimal elements and put M = P − Q. Assume that for all m ∈ M, P_{>m} is n-connected. Then

(1) P_{Q} is (n + 1)-connected.

(2) If Q is k-connected, for k ≤ n + 1, then P is k-connected.

Proof. Since P_{Q} ≃ ⋁_{m∈M} ΣP_{>m}, and since the latter complex is (n + 1)-connected, so is P_{Q}. Then (2) follows from 2.6.

3. The proof of the Main Theorem

We again work in the abstract setting of a poset P which is an extension of Q, and suppose further that we are given a subset I ⊆ Q such that I is an ideal in P. We will say that I has the join property with respect to Q (written in short JQ-property) if for each q ∈ Q the set I_{≤q} is non-empty and has a join in I. Thus for each q ∈ Q there is a unique largest member of I less than or equal to q. We will say that I has the strong join property with respect to Q (written in short SJQ-property) if I has the JQ-property and for all m ∈ M = P − Q and i ∈ I, whenever m and i have an upper bound in P (and hence in Q) they have a join in P. In this case for each m ∈ M we denote I_{≤≥m} = {i ∈ I | {m, i} has an upper bound} regarded as a subposet of I. Note that in this definition and in Lemma 3.1 we do not require that the set M (defined to be P − Q) consist only of minimal elements.

Our application of these notions will be in the situation where N ◁ G with p | |N|. We take P = A_{p}(G), Q = {A ∈ A_{p}(G) | A ∩ N ≠ 1}. Then I = A_{p}(N) has the SJQ-property.

(3.1) LEMMA. Suppose that P is a G-poset with G-invariant subsets I ⊆ Q ⊆ P such that P is an extension of Q and I is an ideal in P. Put M = P − Q.

(1) If I has the JQ-property then the map ϕ : Q → I defined by ϕ(q) = join of I_{≤q} is a G-homotopy equivalence with G-homotopy inverse the inclusion map ι : I → Q.

(2) If I has the SJQ-property then for all m ∈ M the map ϕ_{m} : Q_{>m} → I_{≤≥m} defined by ϕ_{m}(q) = ϕ(q) is a G_{m}-homotopy equivalence with G_{m}-homotopy inverse ψ_{m} : I_{≤≥m} → Q_{>m} defined by ψ_{m}(i) = join of {m, i}.

Proof. All of the maps mentioned are equivariant maps of posets, and they satisfy

ϕι = id_{I},
ιϕ(q) ≤ q for all q ∈ Q,
ψ_{m}ϕ_{m}(q) ≤ q for all q ∈ Q_{>m},
ϕ_{m}ψ_{m}(i) ≥ i for all i ∈ I_{≤≥m}.

Hence by [6, 1.1] ι, ϕ and ψ_{m}, ϕ_{m} are pairs of mutually inverse equivariant homotopy equivalences.

We will apply Lemma 3.1 in the situation where P is an extension of Q by minimal elements, in which case P_{>m} = Q_{>m}.

(3.2) THEOREM. Suppose that P is a G-poset which is an extension of Q where Q is G-invariant. Put M = P − Q, and in (2) – (4) assume that M consists of minimal elements. Suppose that I ⊆ Q is a G-invariant ideal of P with the SJQ-property.

(1) There is a long exact sequence of ℤG-modules

⋯ → H̃_{n}(I)⟶^{ι_{∗}}H̃_{n}(P)⟶^{κ_{∗}}H̃_{n}(P_{Q})⟶^{s}H̃_{n−1}(I) → ⋯

where ι_{∗}, κ_{∗} are the maps on homology induced by the obvious inclusion maps ι, κ.

(2) P_{Q} ≃_{G} ⋁_{m∈M} ΣI_{≤≥m}.

(3) For all n ≥ 0

\begin{aligned}H̃_{n}(P_{Q}) & ≅ ⨁_{m∈M} H̃_{n−1}(I_{≤≥m}) \\ & ≅ ⨁_{m∈[G\M]} H̃_{n−1}(I_{≤≥m}) ↑_{G_{m}}^{G} \end{aligned}

as ℤG-modules.

(4) With respect to the first isomorphism of (3) the map s of (1) is given by

⨁_{m∈M}(ι_{m})_{∗}

where ι_{m} : I_{≤≥m} → I is the inclusion map.

Proof. (1) We identify the terms in the long exact sequence of 2.5 using the homology isomorphisms which we deduce from 3.1. It may help to consider the following diagram in which the top row is the sequence of 2.5 and all vertical arrows are isomorphsms, μ having been defined in 2.4.

⋯ ⟶ H̃_{n}(Q) ⟶^{ι_{∗}}  H̃_{n}(P) ⟶^{κ_{∗}}  H̃_{n}(P_{Q})  ⟶^{r} H̃_{n−1}(Q)
μ↑  ^{ϕ_{∗}}↓↑^{ι_{∗}}
⨁ H̃_{n−1}(P_{>m})  H̃_{n−1}(I)
⊕ϕ_{m∗}↓↑^{⊕ψ_{m∗}}
⨁ H̃_{n−1}(I_{≤≥m})

The sequence presented here thus has s = ϕ_{∗}r.

(2) The homotopy equivalences ψ_{m} yield a G-homotopy equivalence

∨Σψ_{m} : ⋁_{m∈M} ΣI_{≤≥m} → ⋁_{m∈M} ΣP_{>m},

noting here that P_{>m} = Q_{>m}. Combining this with 2.5(2) gives the result.

(3) The first isomorphism is the composite

⨁_{m∈M} H̃_{n−1}(I_{≤≥m}) ⟶^{⊕ψ_{m∗}} ⨁_{m∈M} H̃_{n−1}(P_{>m})⟶^{μ}H̃_{n}(P_{Q}),

where in 2.4 we saw that μ is an isomorphism. The argument that this is an induced module was given in 2.3(2).

(4) On considering the diagram presented in the proof of (1) we see that we have to show ϕ_{∗}rμψ_{m∗} = ι_{m∗} where ι_{m} : I_{≤≥m} → I. We saw in 2.5(4) that rμ = ⊕ι_{m∗}, where in that case ι_{m} : P_{>m} → Q. Since ϕ_{m} is the restriction of ϕ to P_{>m} we have ι_{m}ϕ_{m} = ϕι_{m} and so

ϕ_{∗}rμψ_{m∗} = ϕ_{∗}(⊕ι_{m∗})ψ_{m∗}
= ϕ_{∗}ι_{m∗}ψ_{m∗}
= ι_{m∗}ϕ_{m∗}ψ_{m∗}
= ι_{m∗}

as required, since ϕ_{m} and ψ_{m} are mutually inverse homotopy equivalences.

We now translate Theorem 3.2 into the group theoretic situation of the Main Theorem. We first verify separately one of the identifications that will be needed for this.

(3.3) LEMMA. Let G be a finite group, p a prime and N ◁ G. Then

A_{p}(N)_{≤≥A} = A_{p}(C_{N}(A)).

Proof. We have

A_{p}(N)_{≤≥A} = {K ∈ A_{p}(N) | A and K have an upper bound}.

Now A and K have an upper bound precisely if they are subgroups of some elementary abelian subgroup, which happens precisely if A and K centralize each other, i.e. K ≤ C_{N}(A).

Proof of the Main Theorem. Let G, p and N be as in the main theorem. Let P = A_{p}(G), I = A_{p}(N) and Q = {A ∈ P | A∩N ≠ 1}. Notice that P_{Q} = A_{p}(G)_{N}. It is almost immediate that P, Q and I satisfy all the hypotheses of Theorem 3.2, and of course P is a G-poset under conjugation and Q, I are G-invariant. The Main Theorem is the statement of Theorem 3.2 in this particular case.

We check two things in particular. If A ∈ P − Q then A is an elementary abelian p-group with A ∩ N = 1, so by hypothesis A has order p. Thus P is an extension of Q by minimal elements and P − Q = M as defined in the statement of the Main Theorem. Secondly, for A ∈ M we have I_{≤≥A} = A_{p}(C_{N}(A)) by Lemma 3.3.

4. The Lefschetz invariant and the generalized Steinberg module

As far as computing the homology of A_{p}(G) is concerned, we were able in our Main Theorem to give a sharp description of A_{p}(G)_{N} only in the case when all elementary abelian p-subgroups A ≤ G satisfying A ∩ N = 1 are cyclic. If we weaken the type of information we try to compute and do not ask for the full homology, we may give further results without the strong hypothesis on N.

Following [5], if Δ is a G-simplicial complex we shall use the notation

Λ̃(Δ) = ∑_{i=−1}^{dim Δ}(−1)^{i}Δ_{i}

L̃(Δ) = ∑_{i=−1}^{dim Δ}(−1)^{i}C_{i}(Δ)

respectively for the reduced Lefschetz invariant of Δ in the Burnside ring of G, and the reduced Lefschetz module of Δ in the Green ring of RG-modules. Here Δ_{i} is the G-set of isimplices of Δ, and C_{i}(Δ) is the i-dimensional chain group of Δ, taken over a commutative coefficient ring R. In order to make sense of the Green ring of finitely generated RGmodules we will suppose that R is either a field or a complete discrete valuation ring. We take Δ_{−1} to be a single point with trivial G-action, and C_{−1}(Δ) = R the trivial module. As usual, in case Δ = Δ(P) arises from a poset P, we will write simply Λ̃(P) and L̃(P).

The generalized Steinberg module of the finite group G at the prime p is defined to be

St_{p}(G) = L̃(A_{p}(G))

as an element of the Green ring of RG-modules. A survey of some of its properties can be found in [7]. There is a homomorphism from the Burnside ring to the Green ring which takes every G-set to the corresponding permutation module, and evidently St_{p}(G) is the image of Λ̃(A_{p}(G)) under this homomorphism. We will prove our identities for St_{p}(G) by first establishing the corresponding identities for Λ̃(A_{p}(G)) and then applying this homomorphism.

We are about to invoke a theorem of Thévenaz which applies to posets of the form A +_{C} B where A and B are posets and C is an ideal in the poset A × B. The poset A +_{C} B is defined to be the poset whose underlying set is the disjoint union of A and B, and whose ordering is defined by the ordering of A, the opposite ordering of B and the extra condition:

if a ∈ A, b ∈ B and (a, b) ∈ C then a < b.

If A and B are G-posets and if C is a G-invariant ideal then A +_{C} B is again a G-poset.

In such a poset, A is an ideal of A +_{C} B, and B is an ideal of the opposite poset. Conversely, in every poset P with a subset B which is an ideal in P^{op}, the complementary set A = P −B is an ideal in P, and P = A+_{C}B^{op} where C = {(x, y) | x ∈ A, y ∈ B, x < y}. Thus Thévenaz’s theorem applies to our situation where P is an extension of Q and M = P − Q, so that P = M +_{C} Q^{op}.

(4.1) THEOREM. Suppose that P is a G-poset which is an extension of Q where Q is G-invariant. Put M = P − Q. (1)

Λ̃(P) = Λ̃(Q) + ∑_{m∈[G\M]}(Λ̃(M_{<m}) · Λ̃(Q_{>m})) ↑_{G_{m}}^{G} .

(2) Suppose further that I is a G-invariant ideal of P with I ⊆ Q, and that I has the SJQ-property. Then

Λ̃(P) = Λ̃(I) + ∑_{m∈[G\M]}(Λ̃(M_{<m}) · Λ̃(I_{≤≥m})) ↑_{G_{m}}^{G} .

Proof. Part (1) is a restatement of the first formula of [5, 3.3] in the present context. Part (2) follows from part (1) and 3.1, since Λ̃ is constant on G-homotopy equivalent complexes [5, 1.3].

(4.2) THEOREM. Suppose that G is a finite group with a normal subgroup N such that p | |N|. Then

Λ̃(A_{p}(G)) = Λ̃(A_{p}(N)) + ∑_{A∈Ap(G), A∩N=1 up to conjugacy}(Λ̃(A_{p}(GL(A))) · Λ̃(A_{p}(C_{N}(A)))) ↑_{N_{G}(A)}^{G}

in the Burnside ring, and so

St_{p}(G) = St_{p}(N) + ∑_{A∈Ap(G), A∩N=1 up to conjugacy}(St_{p}(GL(A)) ⊗ St_{p}(C_{N}(A))) ↑_{N_{G}(A)}^{G}

in the Green ring.

Proof. As in the proof of the Main Theorem in Section 3 we take P = A_{p}(G), I = A_{p}(N), Q = {A ∈ P | A ∩ N ≠ 1} and M = {A ∈ P | A ∩ N = 1}. We have seen in Lemma 3.3 that I_{≤≥A} = A_{p}(C_{N}(A)) in this situation, and it is clear that M_{<A} is the poset of proper subspaces of A. This is in fact the poset which defines the building of GL(A), and so we write St_{p}(GL(A)) for its Lefschetz module in the final equation.

The most straightforward (non-trivial) situation in which we may apply the formula for the Steinberg module in 4.2 occurs when G satisfies the condition that A ∩ N = 1, A an elementary abelian subgroup of G implies A is cyclic. In this case, if A ≅ C_{p} then A has no proper subspaces and so St_{p}(GL(A)) = −R. Thus

St_{p}(G) = St_{p}(N) − ∑_{A∈Ap(G), A∩N=1 up to conjugacy} St_{p}(C_{N}(A)) ↑_{N_{G}(A)}^{G} .

This formula may also be deduced by taking the alternating sum of the terms in the long exact sequence of the Main Theorem. It seems that this formula is known to Bouc [3] who uses it to relate St_{2}(S_{n}) and St_{2}(A_{n}).

5. Examples

We present two examples to illustrate the theory we have developed. In the first we consider the situation A_{n}◁S_{n}, and only state the result when n ≥ 6 to avoid the exceptional small cases which may easily be worked by these and other methods.

(5.1) PROPOSITION. For each n ≥ 6 there is a long exact sequence

⋯ ⟶ H̃_{s}(A_{2}A_{n}) ⟶ H̃_{s}(A_{2}S_{n}) ⟶ H̃_{s−1}(A_{2}S_{n−2}) ↑_{S_{2}×S_{n−2}}^{S_{n}}
⟶ H̃_{s−1}(A_{2}A_{n}) ⟶ H̃_{s−1}(A_{2}S_{n}) ⟶ H̃_{s−2}(A_{2}S_{n−2}) ↑_{S_{2}×S_{n−2}}^{S_{n}}
⋮  ⋮  ⋮
⋯ ⟶ H̃_{1}(A_{2}A_{n}) ⟶ H̃_{1}(A_{2}S_{n}) ⟶  0.

Proof. We apply the Main Theorem, in which the conjugacy classes of the set M are represented by the groups ⟨(1, 2)⟩, ⟨(1, 2)(3, 4)(5, 6)⟩, ⟨(1, 2)(3, 4)(5, 6)(7, 8)(9, 10)⟩, … Let us write t_{r} = (1, 2)(3, 4) ⋯ (4r + 1, 4r + 2) ∈ S_{n}. Then C_{A_{n}}(t_{r}) contains the subgroup of index 2 in ⟨(1, 2), (3, 4), …, (4r + 1, 4r + 2)⟩ which consists of even permutations, and this is a normal 2-subgroup of C_{A_{n}}(t_{r}). Thus A_{2}(C_{A_{n}}(t_{r})) is contractible, except when r = 0, in which case C_{A_{n}}(1, 2) ≅ S_{n−2}. This gives the general terms of the long exact sequence, and it terminates as indicated because all of the spaces A_{2}(C_{A_{n}}(t_{r})) are connected.

For our second example we consider a Frobenius group C_{q} ⋊ C_{p} where p is prime with p | q −1, and form the wreath product G = (C_{q} ⋊C_{p})≀C_{p}. The base group N = (C_{q} ⋊C_{p})^{p} is a normal subgroup of index p. This example is of interest because of counterexamples provided by Alperin to an incorrect conjecture made earlier by one of us [7, 3.2]. The conjecture was that H̃_{n}(A_{p}(G))⊗ℤ_{p} should always be a projective ℤ_{p}G-module, where ℤ_{p} denotes the p-adic integers, but in [1] Alperin shows that the top non-vanishing homology group of A_{p}(S_{p^{2}}) with p ≥ 5, is not projective. He has also observed that the wreath product group G provides a counterexample if p ≥ 3, using a similar argument to the one of S_{p^{2}}. We will now compute the homology of A_{p}(G).

(5.2) PROPOSITION. Let G = (C_{q} ⋊ C_{p}) ≀ C_{p} where C_{q} ⋊ C_{p} is a Frobenius group, p a prime, and let N = (C_{q} ⋊ C_{p})^{p} be the base group. Put V = H̃_{0}(A_{p}(C_{q} ⋊ C_{p})), the coordinate-sum zero submodule of rank q − 1 in the permutation module on the q Sylow p-subgroups of C_{q} ⋊ C_{p}. If p ≠ 2 then

H̃_{n}(A_{p}(G)) = \{ \begin{matrix}V ↑_{N}^{⊗G} & if n = p − 1, \\ V ↑_{δ(C_{q}⋊C_{p})×C_{p}}^{G} & if n = 1, \\ 0 & otherwise. \end{matrix}

If p = 2 then H̃_{n}(A_{p}(G)) = 0 unless n = 1, in which case there is a short exact sequence

0 → V ↑_{N}^{⊗G}→ H̃_{1}(A_{p}(G)) → V ↑_{δ(C_{q}⋊C_{p})×C_{p}}^{G}→ 0.

To explain the notation, we may regard V as a module for N = (C_{q} ⋊ C_{p})^{p} via projection onto the first C_{q} ⋊ C_{p} factor. Then V ↑_{N}^{⊗G} denotes the tensor induction. We consider also the diagonal subgroup δ(C_{q} ⋊ C_{p}) ⊆ N, and regard V as a module for δ(C_{q} ⋊ C_{p}) × C_{p}, where the C_{p} factor performs the wreath action, via projection onto δ(C_{q} ⋊ C_{p}) ≅ C_{q} ⋊ C_{p}.

Proof. One sees easily that all complements to N in G are conjugate, since any two are conjugate to complements for the base group in C_{p} ≀C_{p} ≅ 𝔽_{p}C_{p} ⋊C_{p} and here complements are conjugate since H^{1}(C_{p}, 𝔽_{p}C_{p}) = 0. The centralizer in N of such a complement is δ(C_{q} ⋊ C_{p}) and so the long exact sequence of the Main Theorem takes the form

⋯ → H̃_{n}(A_{p}(N)) → H̃_{n}(A_{p}(G)) → H̃_{n−1}(A_{p}(C_{q} ⋊ C_{p})) ↑_{δ(C_{q}⋊C_{p})×C_{p}}^{G}→ ⋯

Now A_{p}(C_{q} ⋊ C_{p}) is the set of q Sylow p-subgroups of C_{q} ⋊ C_{p} and so its homology is zero except in dimension zero, where it is V. As for A_{p}(N), this is homotopy equivalent to the join A_{p}(C_{q} ⋊ C_{p}) ∗ ⋯ ∗ A_{p}(C_{q} ⋊ C_{p}) with p factors by [4], and using the methods of [6] it is easy to see that Quillen’s homotopy equivalence is equivariant for the action of G with C_{p} permuting the factors. Since A_{p}(C_{q} ⋊ C_{p}) is just a set of q points, or in other words a wedge of (q − 1) 0-spheres, the join of p copies of this space is a wedge of p(q − 1) (p−1)-spheres, in which each C_{q} ⋊C_{p} factor of N permutes the corresponding set of (q −1) of these spheres. Thus we have

H̃_{n}(A_{p}(N)) = \{ \begin{matrix}V ⊗ ⋯ ⊗ V & if n = p − 1, \\ 0 & otherwise. \end{matrix}

In the action on this homology the wreathing C_{p} permutes the factors V in the tensor product and so

H̃_{p−1}(A_{p}(N)) = V ↑_{N}^{⊗G} .

Substituting these identifications into the long exact sequence completes the proof of this result.

(5.3) COROLLARY (Alperin). Let G be the group specified in 5.2. When p ≥ 3, H_{p−1}(A_{p}(G)) ⊗ ℤ_{p} is not a projective ℤ_{p}G-module.

Proof. As a module for the wreathing C_{p}, the tensor induced module V ↑_{N}^{⊗G} is a permutation module on tensors v_{i_{1}} ⊗⋯⊗v_{i_{p}}, where v_{1}, …, v_{q−1} is some basis of V. It has as a direct summand the trivial module, spanned by v_{1} ⊗⋯⊗v_{1}, which is not projective.

Acknowledgement

The authors are grateful to the Department of Mathematics at the California Institute of Technology for the hospitality afforded them. In the case of Y. Segev this extended to sabbatical support during the writing of this paper.

REFERENCES

[1] J.L. Alperin, A Lie approach to finite groups, pp. 1-9 in L.G. Kovács (ed.), Groups – Canberra 1989, Lecture Notes in Math. 1456, Springer 1990.

[2] M. Aschbacher and Y. Segev, Locally connected simplicial maps, Israel J. Math. 77 (1992), 285-303.

[3] S. Bouc, Exponentielle et modules de Steinberg, preprint.

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

[5] J. Thévenaz, Permutation representations arising from simplicial complexes, J. Combinatorial Theory Ser. A 46 (1987), 121-155.

[6] J. Thévenaz and P.J. Webb, Homotopy equivalence of posets with a group action, J. Combinatorial Theory Ser. A 56 (1991), 173-181.

[7] P.J. Webb, Subgroup complexes, pp. 349-365 in P. Fong (ed.), The Arcata Conference on Representations of Finite Groups, AMS Proceedings of Symposia in Pure Mathematics 47 (1987).

Department of Mathematics Ben – Gurion University Beer – Sheva 84105 Israel Department of Mathematics University of Minnesota Minneapolis Minnesota 55455, USA