THE EULER CLASS OF A SUBSET COMPLEX

ASLI GÜÇLÜKAN AND ERGÜN YALÇIN

Abstract. The subset complex Δ(G) of a finite group G is defined as the simplicial complex whose simplices are nonempty subsets of G. The oriented chain complex of Δ(G) gives a ℤG-module extension of ℤ by ℤ̃ where ℤ̃ is a copy of integers on which G acts via the sign representation of the regular representation. The extension class ζ_{G} ∈ Ext_{ℤG}^{|G|−1}(ℤ, ℤ̃) of this extension is called the Ext class or the Euler class of the subset complex Δ(G). This class was first introduced by Reiner and Webb [15] who also raised the following question: What are the finite groups for which ζ_{G} is nonzero?

In this paper, we answer this question completely. We show that ζ_{G} is nonzero if and only if G is an elementary abelian p-group or G is isomorphic to ℤ/9, ℤ/4 × ℤ/4, or (ℤ/2)^{n} ×ℤ/4 for some integer n ≥ 0. We obtain this result by first showing that ζ_{G} is zero when G is a nonabelian group, then by calculating ζ_{G} for specific abelian groups. The key ingredient in the proof is an observation by Mandell which says that the Ext class of the subset complex Δ(G) is equal to the (twisted) Euler class of the augmentation module of the regular representation of G.

We also give some applications of our results to group cohomology, to filtrations of modules, and to the existence of Borsuk-Ulam type theorems.

1. Introduction

Let G be a finite group and let X be a finite G-set, say of order n+1. The subset complex of X is defined as the simplicial complex Δ(X) whose simplices are nonempty subsets of X. One can choose an orientation on Δ(X) by choosing an ordering x_{0} < ⋯ < x_{n} for elements of X. The oriented chain complex of Δ(X) augmented by a copy of the trivial module gives an exact sequence of ℤG-modules

ε_{X} : 0 → ℤ̃ → C_{n−1}(Δ(X)) → ⋯ → C_{0}(Δ(X)) → ℤ → 0

where ℤ̃ is a copy of the trivial module on which G acts via the sign representation of X. Associated to this exact sequence, there is an extension class ζ_{X} ∈ Ext_{ℤG}^{n}(ℤ, ℤ̃). This class was first introduced by Reiner and Webb [15] and is called the Ext class of the subset complex of X.

In [15], it is shown that the extension class ζ_{X} is an essential class when X = G/1 is the transitive G-set with a point stabilizer. In this case, Δ(X) is the subset complex of the group G, and we denote the associated Ext class by ζ_{G}. Note that if ζ_{G} is a nontrivial class, then this means that the group G has nontrivial essential cohomology. On the other hand, if ζ_{G} is zero, then the hypercohomology spectral sequence associated to the subset complex collapses at the E_{2}-page. Because of these consequences, Reiner and Webb asked for which finite groups ζ_{G} ≠ 0.

The first author is supported by Tübitak PhD scholarship, the second author is supported by TÜBİTAK-BDP and by TÜBA-GEBİP/2005-16.

In this paper, we give a complete answer to this question. Our starting point is the observation that the extension class ζ_{G} is, in fact, the (twisted) Euler class of the augmentation module I_{G} = ker{ℝG → ℝ}. In their paper, Reiner and Webb [15] attribute this observation to Mandell, but they do not provide a proof. Since our arguments are based on this observation, we give a proof of this fact in Section 2. Because of this observation, we sometimes call the Ext class ζ_{G}, the Euler class of the subset complex to emphasize that it is an Euler class.

It has been shown in [15] that ζ_{G} is zero when G is a composite group. Therefore throughout the paper we consider only p-groups. The main result of the paper is the following:

Theorem 1.1. If G is a finite nonabelian group, then ζ_{G} = 0.

The proof follows from a reduction argument. We show that if ζ_{G} is nonzero then ζ_{H/K} is nonzero for every subquotient H/K of G. Thus the minimal counterexample to Theorem 1.1 should have all proper subquotients abelian. It is easy to classify nonabelian p-groups whose proper subquotients are all abelian. Such p-groups are either of order p^{3} or isomorphic to the modular group M_{p^{k}} with k ≥ 4. By direct calculation, we show that the Euler classes of the augmentation modules of these groups are all zero. This proves Theorem 1.1.

In the rest of the paper, we consider abelian p-groups. First we show that ζ_{G} is zero when G is isomorphic to ℤ/8 or (ℤ/4)^{2} ×ℤ/2, and hence conclude that any group which has a subquotient isomorphic to one of these groups has zero Euler class. This shows that if G is a 2-group with ζ_{G} ≠ 0, then G is either elementary abelian or is isomorphic to ℤ/4 × ℤ/4 or (ℤ/2)^{n} × ℤ/4 for some n ≥ 0. Then, we show that the Euler class ζ_{G} for these groups is nonzero. For p > 2, we use similar arguments and obtain that ζ_{G} is nonzero if and only if G is elementary abelian or isomorphic to ℤ/9. Hence, we conclude the following:

Theorem 1.2. Let G be a finite abelian group. Then, ζ_{G} is nonzero if and only if G is either an elementary abelian p-group or is isomorphic to ℤ/9, ℤ/4 × ℤ/4, or (ℤ/2)^{n} × ℤ/4 for some integer n ≥ 0.

There are many consequences of Theorems 1.1 and 1.2. In section 6, we discuss some immediate consequences such as the classification of p-groups for which the mod p-reduction of ζ_{G} is zero. We also give a complete list of p-groups which has (ζ_{G})^{2} ≠ 0. In Section 7, we give some applications of our results to filtrations of modules. Another application of our results is to the existence of Borsuk-Ulam type theorems which are in turn related to the Tverberg problem in combinatorics. We discuss these in Section 8.

In section 9, we consider the hypercohomology spectral sequences associated to the boundary of the subset complex. When G is not one of the groups listed in Theorem 1.2, the cohomology spectral sequence collapses at the E_{2}-page. In particular, the edge homomorphism H_{G}^{∗}(pt, ℤ) → H_{G}^{∗}(∂Δ(G), ℤ) induced by the constant map ∂Δ(G) → pt is injective. Since the G-complex ∂Δ(G) has no fixed points, this gives, for example, that the essential ideal is nilpotent with nilpotency degree ≤ |G|. The collapsing of the cohomology spectral sequence also allows one to look at the isotropy spectral sequence to obtain information about the integral cohomology of G.

In Section 10, we consider the barycentric subdivision of the subset complex. The reason for introducing the barycentric subdivision is that as a G-complex, the subset complex is not admissible. This causes problems when we want to calculate the product structure of the cohomology ring using the hypercohomology spectral sequence associated to the subset complex. Our main result in this section is a formula for the permutation modules appearing on the chain complex of the barycentric subdivision. We use a power series approach which was introduced by Webb in [21] and later used in [15].

2. The twisted Euler class of a real representation

The main purpose of this section is to show that the Ext class ζ_{X} is equal to the twisted Euler class of the augmentation module I_{X} = ker{ℝX → ℝ} after suitable identifications. By twisted Euler class, we mean the Euler class of a not necessarily orientable real vector bundle. We first discuss the notion of the twisted Euler class of a real representation.

Given a real representation V of a finite group G, we can form a real vector bundle EG ×_{G} V → BG using the Borel construction. The Euler class of V is defined as the Euler class of this vector bundle. In general this bundle is not orientable, but there is a notion of orientation for nonorientable bundles and depending on the choice of orientation one defines a twisted Euler class in a similar way the usual Euler class is defined. Although the definition and the properties of a twisted Euler class are standard, it is hard to find in the literature. A short note on the twisted Euler class of a real representation is given in the appendix of [7]. We include some of this material here for the convenience of the reader and also to introduce the notation.

Let B be a connected (pointed) CW -complex and ξ : V → E ⟶^{π} B be an n-dimensional real vector bundle over B. The Stiefel-Whitney class w_{1}(ξ) ∈ H^{1}(B, ℤ/2) can be considered as a homomorphism π_{1}(B) → {±1}. Let ℤ(ξ) denote the one dimensional integral representation of π_{1}(B) where the action is given by w_{1}(ξ). When w_{1}(ξ) is trivial the bundle is called orientable. If w_{1}(ξ) is not trivial, then we say the bundle is non-orientable. In this case, H^{n}(π^{−1}(b_{0}), π^{−1}(b_{0})\{0}) is isomorphic to ℤ(ξ) as a π_{1}(B)-module. An orientation of ξ is defined as a π_{1}(B)-module isomorphism

ω : H^{n}(π^{−1}(b_{0}), π^{−1}(b_{0})\{0}) → ℤ(ξ).

For each vector bundle, there are two orientations, ω and −ω, which are opposite to each other. A bundle ξ on which the orientation ω is fixed, is denoted by ξ^{ω}. We sometimes call such a bundle oriented although it may not be an orientable bundle in the above sense. For an oriented bundle ξ^{ω}, the Thom isomorphism

ϕ_{L} : H^{r}(B, L) → H^{r+n}(E, E_{0}; L ⊗ ℤ(ξ))

is given by ϕ_{L}(x) = π^{∗}(x) · U where U is the Thom class and L is a coefficient bundle over B. Here the map π^{∗} : H^{∗}(B, L) → H^{∗}(E, L) is induced from the projection π : E → B. The Euler class of ξ^{ω} is defined by

e(ξ^{ω}) = ϕ_{ℤ(ξ)}^{−1}(U^{2}) ∈ H^{n}(B, ℤ(ξ)).

Sometimes this Euler class is referred to as the twisted Euler class to emphasize that the coefficients are twisted, but we will not make this distinction here. The following are some basic properties of the Euler class.

Lemma 2.1. Let ξ^{ω} and η^{ν} be two oriented real vector bundles of dimension n and m respectively. Then,

1) e(ξ^{ω}) = −e(ξ^{−ω}).

2) The mod 2 reduction of e(ξ^{ω}) is equal to the top Stiefel-Whitney class of ξ.

3) e(ξ^{ω})e(η^{ν}) = e((ξ ⊕ η)^{ω⊕ν}).

Proof. The proofs of these statements are standard and can be found, for example, in [11]. Note that in the third statement to get the equality the cup product on the left is defined as the composition

H^{n}(B, ℤ(ω)) ⊗ H^{m}(B, ℤ(ν)) → H^{n+m}(B, ℤ(ω) ⊗ ℤ(ν)) → H^{n+m}(B, ℤ(ω ⊕ ν))

where the second map is induced by the canonical isomorphism ℤ(ξ) ⊗ ℤ(η) → ℤ(ξ ⊕ η) and ω ⊕ ν is the orientation of ξ ⊕ η given by w_{1}(ξ) + w_{1}(η). □

For a real representation V of G, the first Stiefel-Whitney class of V is defined as the first Stiefel-Whitney class of the bundle ξ_{V} : EG ×_{G} V → BG and is given by the composition

sgn(V) : G → GL(V) ⟶^{det} ℝ^{×}.

So, the associated bundle ξ_{V} is orientable if and only if G action preserves a chosen orientation of V. Note that choosing an orientation ω : H^{n}(V, V \{0}) → ℤ(ξ_{V}) for the bundle ξ_{V} is the same as choosing an orientation for V. If V is a real representation with a fixed orientation, then we denote the module ℤ(ξ_{V}) by ℤ̃, and the associated Euler class by e(V).

The Euler class can also be defined as the first obstruction to the existence of a nonzero section. Let us choose an arbitrary G-invariant inner product on V and let S(V) be the set of all unit vectors in V with respect to this inner product. Associated to the G-space S(V), there is a sphere bundle EG ×_{G} S(V) → BG with fibers S(V).

Lemma 2.2. Let V be an n-dimensional real representation of G with a fixed orientation. The Euler class e(V) ∈ H^{n}(BG, ℤ̃) is the first obstruction for an existence of a section for the sphere bundle EG ×_{G} S(V) → BG associated to S(V). Equivalently, the Euler class e(V) is the first obstruction o_{n} ∈ H_{G}^{n}(EG, ℤ̃) for finding a G-map f : EG → S(V).

Proof. See Milnor and Stasheff [11]. □

Given a finite G-CW -complex Y which has the homology of a sphere, say of dimension n − 1, there is a concept of polarization which is commonly used to fix a k-invariant for the complex. A polarization of Y is a pair of isomorphisms φ : H_{0}(Y) → ℤ and ψ : H_{n−1}(Y) → ℤ̃, and associated to each polarization, there is a unique k-invariant ζ ∈ H^{n}(BG, ℤ̃) defined as follows: Note that given a polarized G-CW -complex Y with polarizations φ and ψ, we get an extension of ℤG-modules of the form

ε_{V} : 0 → ℤ̃ → C_{n−1}(Y) → ⋯ → C_{0}(Y) → ℤ → 0

using the polarizations at the ends of the extension to get ℤ and ℤ̃. This defines a unique extension class ζ(Y, φ, ψ) ∈ Ext_{ℤG}^{n}(ℤ, ℤ̃). The k-invariant is defined as the corresponding class in H^{n}(BG, ℤ̃).

It is easy to see that if one fixes the polarization φ : H_{0}(Y) → ℤ to be the one given by augmentation map C_{0}(Y) → ℤ, then the choice of the second polarization corresponds to the choice of orientation for the chain complex. In the case of the unit sphere S(V) of a real representation V, the polarization ψ : H_{n−1}(S(V)) → ℤ corresponds to an orientation of V. If V comes with a fixed orientation, then there is a unique extension class ζ_{V} ∈ Ext_{ℤG}^{n}(ℤ, ℤ̃) associated to S(V). The following is a standard result known by experts in the field.

Proposition 2.3. Let V be an n-dimensional real representation of G with a fixed orientation. Let ℤ̃ denote the one-dimensional integral representation induced from the sign representation of V. Consider the exact sequence

ε_{V} : 0 → ℤ̃ → C_{n−1}(S(V)) → ⋯ → C_{0}(S(V)) → ℤ → 0

obtained by applying the polarizations coming from the fixed orientation of V. Let ζ_{V} ∈ Ext_{ℤG}^{n}(ℤ, ℤ̃) be the extension class of this extension. Then, the image of ζ_{V} is equal to the Euler class e(V) under the canonical isomorphism Ext_{ℤG}^{n}(ℤ, ℤ̃) ≅ H^{n}(G, ℤ̃).

Proof. Since π_{i}(S(V)) = 0 for i ≤ n − 1, we can construct a G-map f_{n−1} : EG^{(n−1)} → S(V) in such a way that the induced map on 0-th homology is identity. By Lemma 2.2, the obstruction to extending f_{n−1} to a G-map f_{n} : EG^{(n)} → S(V) is the Euler class e(V). By obstruction theory, this obstruction class is represented by a cocycle in Hom_{G}(C_{n}(EG), H_{n−1}(S(V)) which is defined by the composition

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} o_{n}

It is clear from this diagram that o_{n} is the lifting of the identity, so e(V) corresponds to the extension class of the bottom extension under the isomorphism H^{n}(G, H_{n−1}(S(V))) ≅ Ext_{ℤG}^{n}(ℤ, H_{n−1}(S(V))). Note that since V has a fixed orientation, there is a canonical isomorphism H_{n−1}(S(V), ℤ) ≅ ℤ̃ which we can use to replace H_{n−1}(S(V), ℤ) with ℤ̃ in the above argument. So, e(V) corresponds to ζ_{V} under the isomorphism Ext_{ℤG}^{n}(ℤ, ℤ̃) ≅ H^{n}(G, ℤ̃). □

Now, we are ready to show that the extension class ζ_{X} is the same as the Euler class of the augmentation module I_{X} = ker{ℝX → ℝ}. Recall that in the introduction we defined ζ_{X} as the extension class of the extension

ε_{X} : 0 → ℤ̃ → C_{n−1}(Δ(X)) → . . . · · · → C_{0}(Δ(X)) → ℤ → 0

where Δ(X) is the subset complex of X. It is easy to see that this extension is equivalent to the extension

0 → H_{n−1}(∂Δ(X)) → C_{n−1}(∂Δ(X)) → ⋯ → C_{0}(∂Δ(X)) → ℤ → 0

where ∂Δ(X) denotes the boundary of the subset complex Δ(X). For the boundary of the subset complex, we have the following observation.

Lemma 2.4 ([2], Lemma 2.2). Let G be a finite group and X be a finite G-set. Suppose S(I_{X}) denotes the unit sphere of the augmentation ideal I_{X} = ker{ℝX → ℝ} and |∂Δ(X)| denotes the realization of the boundary of the subset complex Δ(X). Then, there is a Ghomeomorphism between the topological spaces S(I_{X}) and |∂Δ(X)|.

Proof. Let {x_{0}, …, x_{n}} be the set of elements of X. We can regard I_{X} as the normal space of the vector (1, …, 1) and x_{i} as the i-th unit vector. Let v_{i} be the unit vector of the projection of x_{i} into I_{X}. Then the set {v_{0}, …, v_{n}} is an affinely independent set of vectors in S(I_{X}). Let Δ′(X) be the n-simplex with vertex set {v_{0}, …, v_{n}}. Let us define a map ϕ : Δ(X) → Δ′(X) by ϕ(x_{i}) = v_{i}. It is easy to see that ϕ is a G-homeomorphism and it sends ∂Δ(X) to the boundary of Δ′(X), hence it induces a G-homeomorphism between the associated topological spaces S(I_{X}) and |∂Δ(X)|. □

Now, we are ready to prove our main theorem in this section.

Theorem 2.5. Let G be a finite group and X be a finite G-set. Then the Ext class ζ_{X} is equal to the Euler class e(I_{X}) of the augmentation module I_{X} under the canonical isomorphism Ext_{ℤG}^{n}(ℤ, ℤ̃) ≅ H^{n}(G, ℤ̃).

Proof. Fix an ordering of elements in X so that we have a fixed orientation throughout. By Lemma 2.4, the chain complexes of ∂Δ(X) and S(I_{X}) are chain homotopic. This means that the extension class ζ_{X} also represents the following exact sequence

0 → ℤ̃ → C_{n−1}(S(I_{X})) → ⋯ → C_{0}(S(I_{X})) → ℤ → 0.

However by Proposition 2.3, this extension is represented by the Euler class e(I_{X}). Therefore, the image of ζ_{X} is equal to e(I_{X}) under the isomorphism Ext_{ℤG}^{n}(ℤ, ℤ̃) ≅ H^{n}(G, ℤ̃). □

The above theorem reduces Reiner and Webb’s question to a question about the Euler class of the augmentation module. Since the augmentation module decomposes to irreducible real representations, this observation makes it much easier to calculate the Ext class of the subset complex. In fact, from now on we will call the Ext class, the Euler class of the subset complex to emphasize on the fact that it is actually an Euler class.

Finally, we would like to note that, although in our calculations we use the group cohomology with twisted coefficients, the Euler class itself often lies in the integral cohomology of G (with no twisting). In fact, Reiner and Webb showed that ζ_{G} lies in the cohomology with twisted coefficients if and only if G has a nontrivial cyclic Sylow 2-subgroup (see Lemma 5.4, [15]). In particular, if G is a p-group, then the Euler class is twisted only when G is a cyclic 2-group in which case the coefficients are given by the unique nontrivial map G → ℤ/2.

3. Proof of Theorem 1.1

The key ingredient in the proof of the Theorem 1.1 is the following reduction argument.

Proposition 3.1. If the Euler class ζ_{G} is nonzero, then the Euler class ζ_{H/K} is nonzero for every subquotient H/K of G.

Proof. In [15], it is shown that for any subgroups K ≤ H ≤ G, the extension class ζ_{G/K} is equivalent to the cup product of ζ_{G/H} and N_{H}^{G}(ζ_{H/K}) where N denotes the Evens’ norm map (see Proposition 7.13 in [15]). Therefore, for subgroups 1 ≤ K ≤ H ≤ G we have

ζ_{G} = ζ_{G/H} · N_{H}^{G}(ζ_{H/K} · N_{K}^{H}(ζ_{K})).

From this it follows immediately that if ζ_{H/K} = 0 for some subquotient H/K of G, then ζ_{G} = 0. □

Proposition 3.1 implies that a minimal counterexample to Theorem 1.1 must be a nonabelian p-group whose proper subquotients are all abelian. First we classify such groups and then we show that ζ_{G} = 0 for all groups in the list. This means that there can not be any counterexamples to Theorem 1.1, hence it completes the proof. The classification of all nonabelian p-groups whose proper subquotients are all abelian is given as follows:

Proposition 3.2. Let G be a nonabelian p-group whose proper subquotients are all abelian. Then either G has order p^{3} or G is isomorphic to the modular p-group M_{p^{k}} for some k ≥ 4.

Proof. Since every group of order p^{2} is abelian, nonabelian groups of order p^{3} obviously satisfy the assumption of the theorem. So, let us assume that G is a nonabelian p-group of order |G| > p^{3} whose proper subquotients are all abelian. Let c be a central element of order p in G. Since G/⟨c⟩ is abelian and G is nonabelian, ⟨c⟩ is the commutator group of G. Similarly, any central subgroup of order p is the commutator group and hence G has only one central subgroup of order p. This means that the center Z(G) of G is cyclic. Note that G has an element of order p which is not central because otherwise G has a unique subgroup of order p which implies that G is either cyclic or a generalized quaternion group (see Theorem 4.3 in [6]). But these groups do not satisfy our starting assumption. So, G has an element of order p which is not central, say a. Let s be an element of G which does not commute with a. Since the subgroup generated by s and a is nonabelian, we must have G = ⟨a, s⟩. Note that asa^{−1}s^{−1} = c^{t} for some t ≢ 0 mod p. This gives as^{p}a^{−1} = c^{tp}s^{p} = s^{p}, so s^{p} is central in G. This forces the Frattini subgroup of G to be the subgroup generated by c and s^{p}. Thus, the Frattini subgroup is central, and hence cyclic. Since |G| > p^{3}, the element s^{p} cannot be trivial. So, we have c^{r} = s^{p^{k−2}} for some r ≢ 0 mod p where p^{k−1} is the order of s. Note that we can also assume r = t = 1 by replacing a and c with appropriate powers of themselves. Therefore, G has a presentation

G = ⟨a, s | a^{p} = s^{p^{k−1}} = 1, asa^{−1} = s^{p^{k−2}+1}⟩.

Hence, it is isomorphic to the modular group of order p^{k} with k ≥ 4. □

In order to prove Theorem 1.1, we need to show that the Euler class ζ_{G} is zero for all the groups listed in Proposition 3.2. We will use different arguments for p is odd and p = 2. Let us first deal with the case where p = 2.

Lemma 3.3. The Euler class ζ_{G} is zero when G ≅ Q_{8} or G ≅ M_{2^{k}} with k ≥ 3.

Proof. In both cases the Frattini subgroup Φ(G) of G is cyclic and central, and the quotient G/Φ(G) is isomorphic to the elementary abelian group of order 4. Therefore G has a central extension of the form

0 → Φ(G) → G → ℤ/2 × ℤ/2 → 0.

Let us consider the Lyndon-Hochschild-Serre spectral sequence corresponding to this extension. By Proposition 7.2 in [22], the generator μ of the group H^{3}(ℤ/2 × ℤ/2, ℤ) ≅ ℤ/2 is in the image of the differential d_{3}. This means that inf_{G/Φ(G)}^{G}(μ) is zero in H^{3}(G, ℤ).

The mod 2 reduction of μ is x_{1}x_{2}(x_{1} + x_{2}) which is the top Stiefel-Whitney class of I_{G/Φ(G)}. Since the mod 2 reduction is injective for elementary abelian groups, this gives μ = e(I_{G/Φ(G)}). Therefore Inf_{G/Φ(G)}^{G}ζ_{G/Φ(G)} = 0, and hence ζ_{G} = 0. □

Now, we consider the case p > 2. The modular p-group M_{p^{k}} with k ≥ 4 has an abelian subgroup isomorphic to ℤ/p^{2} × ℤ/p. Thus, the fact that ζ_{G} = 0 for G = M_{p^{k}} with k ≥ 4 is a consequence of the following lemma.

Lemma 3.4. The Euler class ζ_{G} is zero when G ≅ ℤ/p^{2} × ℤ/p and p is odd.

Proof. Let G = ⟨a, b | a^{p^{2}} = b^{p} = 1, ab = ba⟩. Then H^{∗}(G, ℤ) = ℤ[α, β] ⊗ ∧_{p}(χ) where degα = degβ = 2, degχ = 3 and p^{2}α = pβ = pχ = 0 (see [9]). Since the Chern class c_{1} defines an isomorphism Hom(G, ℂ^{×}) ≅ H^{2}(G, ℤ), we can consider the generators α and β of H^{2}(G, ℤ) as the Chern classes of the representations V_{1} : a → ω, b → 1 and V_{2} : a → 1, b → ω^{p} where ω is the primitive p^{2}-th root of unity. With this notation, the Chern class c_{1}(V_{3}) of the one dimensional complex representation V_{3} : a → ω^{p}, b → 1 is equal to pα.

Let W_{2} and W_{3} be the underlying 2-dimensional real representations of V_{2} and V_{3} respectively. Then we have

e(W_{2} ⊕ W_{3}) = e(W_{2})e(W_{3}) = c_{1}(V_{2})c_{1}(V_{3}) = pαβ = 0.

Since W_{2} ⊕ W_{3} is a direct summand of the augmentation module I_{G}, this gives ζ_{G} = 0. □

Now it remains to consider the nonabelian groups of order p^{3} for p > 2. The following lemma solves the problem for this case and hence completes the proof of Theorem 1.1.

Lemma 3.5. Let G be a nonabelian p-group of order p^{3} with p > 2. Then, the Euler class ζ_{G} is zero.

Proof. If G is a p-group of order p^{3} with p > 2, then G is either isomorphic to the extraspecial group E_{p^{3}} of exponent p or to the modular group M_{p^{3}} of exponent p^{2}. In both cases, the exponent of H^{i}(G, ℤ) is p when i is not divisible by 2p (see [9]). Therefore, the mod p reduction map H^{p^{3}−1}(G, ℤ) → H^{p^{3}−1}(G, 𝔽_{p}) is injective. So, it is sufficient to show that the mod p reduction of ζ_{G} is zero.

In the mod p cohomology of the groups E_{p^{3}} and M_{p^{3}}, there are relations of the form xy = 0 and β(x) + xy = 0 respectively, where x, y denote the generators of one dimensional cohomology for each group. If we apply the operator βP β to these relations, then we obtain

β(x) ∏_{j=0}^{p−1} β(jx + y) = 0

This product is a factor of the mod p reduction of ζ_{G}. Hence, ζ_{G} = 0. □

4. Calculations for some abelian 2-groups

In this section we calculate the Euler class ζ_{G} for some small abelian 2-groups. This allows us to narrow the range for the search of abelian 2-groups with nonzero Euler class.

Proposition 4.1. The Euler class ζ_{G} is zero when G ≅ ℤ/8.

Proof. Let H be the maximal subgroup of G. We have ζ_{G} = ζ_{G/H} · e(W) where W is the direct sum of all irreducible two dimensional representations of G. The Euler class ζ_{G/H} is represented by the extension

0 → ℤ̃ → ℤ[G/H] → ℤ → 0.

Consider the long exact sequence associated with this short exact sequence:

⋯ ⟶ H^{6}(G, ℤ̃) ⟶ H^{6}(H, ℤ) ⟶^{tr} H^{6}(G, ℤ) ⟶^{·ζ_{G/H}} H^{7}(G, ℤ̃) ⟶ ⋯
∥  ∥
ℤ/|H|ℤ ⟶^{×2} ℤ/|G|ℤ

It is clear from the above diagram that to show that ζ_{G} = 0, it suffices to show that e(W) ∈ H^{6}(G, ℤ) is divisible by 2. Note that W = W_{1} + W_{2} + W_{3} where W_{i} is the two dimensional real representation such that the action is given by 2πi/8 degree rotation. Also note that if α = e(W_{1}), then e(W_{j}) = jα for all j = 1, 2, 3. This gives e(W) = 6α^{3} which is divisible by 2 as desired. □

The calculation above shows that the Euler class ζ_{G} is zero for all cyclic 2-groups with order greater or equal to 8. An easy calculation shows that the Euler class for ℤ/4 is not zero. We will show later that ζ_{G} is not zero also when G = ℤ/4 × ℤ/4. The next group we consider is G = (ℤ/4)^{2} × ℤ/2. We show that the Euler class for this group is zero. For this calculation, we need the structure of the cohomology of the group ℤ/4 × ℤ/4 with integer coefficients. In [19], Townsley completely describes the integral cohomology of all abelian groups. We quote the result from [19], but since what we need is a very special case of Townsley’s calculations, we provide a proof for the convenience of the reader.

Proposition 4.2 (Townsley [19]). Let G = ℤ/4 × ℤ/4. Then,

H^{∗}(G, ℤ) ≅ ℤ/4[μ_{1}, μ_{2}, μ_{12}]/⟨μ_{12}^{2} − 2μ_{1}μ_{2}(μ_{1} + μ_{2})⟩.

Proof. Let us consider the Lyndon-Hochschield-Serre spectral sequence with E_{2}-page

E_{2}^{p,q} ≅ H^{p}(G/K, H^{q}(K, ℤ)) ⇒ H^{p+q}(G, ℤ)

where K is a cyclic subgroup of G of order 4. Let t_{1} and t_{2} be the generators of H^{∗}(K, ℤ) and H^{∗}(G/K, ℤ). Since H^{2}(G, ℤ) ≅ H^{2}(G/K, ℤ) ⊕ H^{2}(K, ℤ), we have d_{2}(t_{1}) = 0, which implies that d_{2} = 0. By dimension reasons d_{i} = 0 for any i ≥ 2, hence the spectral sequence collapses at E_{2}-page. Thus, the Poincaré series of H^{∗}(G, ℤ) is given by

P_{H^{∗}(G,ℤ)}(t) = \frac{1 + t^{3}}{(1 − t^{2})^{2}} .

Since E_{2}^{1,2} = H^{1}(G/K, H^{2}(K, ℤ)) ≅ ℤ/4, the cohomology ring H^{∗}(G, ℤ) has at least three generators. Let μ_{1} and μ_{2} be the generators of degree 2 and let μ_{12} be the generator of degree 3. Without loss of generality, we can assume res_{K}^{G}μ_{1} = t_{1} and Inf_{G/K}^{G}t_{2} = μ_{2}. We claim that μ_{1} and μ_{2} are algebraically independent. Indeed, if

f(μ_{1}, μ_{2}) = ∑_{i=0}^{k} a_{i}μ_{1}^{i}μ_{2}^{k−i}

is a relation with the smallest degree then the restriction of f(μ_{1}, μ_{2}) to the subgroup K gives a_{k}t_{1}^{k} = 0 and hence a_{k} = 0. Therefore, f(μ_{1}, μ_{2}) = μ_{2}g(μ_{1}, μ_{2}) for some polynomial g(μ_{1}, μ_{2}) with smaller degree. Since μ_{2} is a nonzero divisor, this gives g(μ_{1}, μ_{2}) = 0 which contradicts the minimality of f(μ_{1}, μ_{2}). Therefore, μ_{1} and μ_{2} are algebraically independent as claimed.

Let us assume for the moment that μ_{12}^{2} = 2μ_{1}μ_{2}(μ_{1} + μ_{2}) is the only relation for the generators μ_{1}, μ_{2} and μ_{12}. It implies that

S = ℤ/4[μ_{1}, μ_{2}, μ_{12}]/⟨μ_{12}^{2} − 2μ_{1}μ_{2}(μ_{1} + μ_{2})⟩

is a subring of H^{∗}(G, ℤ). On the other hand, S and H^{∗}(G, ℤ) have the same Poincaré series. So, we obtain H^{∗}(G, ℤ) = S as desired. Now we prove that μ_{12}^{2} = 2μ_{1}μ_{2}(μ_{1} + μ_{2}) is the only relation. Since μ_{12} has an odd degree, we have 2μ_{12}^{2} = 0. This implies that (μ_{12})^{2} = 2f(μ_{1}, μ_{2}) for some polynomial f(μ_{1}, μ_{2}) ∈ H^{6}(G, ℤ). It is easy to show that μ_{12}^{2} is not zero by considering the spectral sequence associated to the extension

0 → ℤ/2 × ℤ/2 → G → ℤ/2 × ℤ/2 → 0.

So, we can assume f(μ_{1}, μ_{2}) = a_{1}μ_{1}^{3} + a_{2}μ_{2}^{3} + a_{3}μ_{1}^{2}μ_{2} + a_{4}μ_{1}μ_{2}^{2} where at least one of the a_{i}’s is nonzero. Since the restriction of μ_{12} to any cyclic subgroup H of G is zero, we get a_{1} = a_{2} = 0 and a_{3} = a_{4} = 1. So, μ_{12}^{2} = 2μ_{1}μ_{2}(μ_{1} +μ_{2}). Suppose now that there is another relation. Then, it must be of the form

μ_{12} · g(μ_{1}, μ_{2}) + h(μ_{1}, μ_{2}) = 0,

but this is impossible since the degree of μ_{1} and μ_{2} are 2 and the degree of μ_{12} is 3. □

Now we are ready to do the following calculation.

Proposition 4.3. The Euler class ζ_{G} is zero when G ≅ (ℤ/4)^{2} × ℤ/2.

Proof. The Euler class ζ_{G} includes inf_{G/Φ(G)}^{G}ζ_{G/Φ(G)} as a factor. So, it is enough to prove that this factor is zero. The cohomology ring H^{∗}(G/Φ(G), ℤ) is generated by the elements u_{I} where the indices I run through the subsets of {1, 2, 3}. The mod 2 reduction of u_{I} is given by the formula

m_{2}(u_{I}) = ( ∏_{i∈I} x_{i})( ∑_{i∈I} x_{i})

where x_{1}, x_{2} and x_{3} are the generators of H^{∗}(G/Φ(G), ℤ/2). By direct calculation, one can show that the mod 2 reduction of u_{1}^{2}u_{23} + u_{2}^{2}u_{13} + u_{3}^{2}u_{12} is equal to the product of one dimensional classes, hence it is equal to the top Stiefel-Whitney class of I_{G/Φ(G)}. Since the reduction modulo 2 is an injective map for elementary abelian groups, we conclude that

ζ_{G/Φ(G)} = u_{1}^{2}u_{23} + u_{2}^{2}u_{13} + u_{3}^{2}u_{12}.

Now we show that the inflation of this element is zero. Let a, b, c be the generators of G with a^{4} = b^{4} = c^{2} = 1. Suppose μ_{1}, μ_{2} and μ_{3} are the generators of H^{2}(G, ℤ) = Hom(G, ℂ^{×}) which are dual to a, b, c respectively. We can choose the generators u_{1}, u_{2}, u_{3} for H^{2}(G/Φ(G), ℤ) in a compatible way and assume that Inf_{G/Φ(G)}^{G}u_{i} = 2μ_{i} for i = 1, 2 and Inf_{G/Φ(G)}^{G}u_{3} = μ_{3}. Since the exponent of H^{∗}(G, ℤ) is 4, we get Inf_{G/Φ(G)}^{G}u_{i}^{2} = 0 for i = 1, 2. Hence

Inf_{G/Φ(G)}^{G}(u_{1}^{2}u_{23} + u_{2}^{2}u_{13} + u_{3}^{2}u_{12}) = μ_{3}^{2} · Inf_{G/Φ(G)}^{G}u_{12}.

On the other hand, we have

Inf_{G/Φ(G)}^{G}u_{12} = InfG ; \overline{G}Inf\overline{G} ; \overline{G}/Φ(\overline{G})u_{12} = InfG ; \overline{G} 2μ_{12} = 2InfG ; \overline{G} μ_{12}

where \overline{G} = G/⟨c⟩ ≅ ℤ/4 × ℤ/4 and μ_{12} is the generator of H^{3}(\overline{G}, ℤ) = ℤ/4. Since 2μ_{3} = 0, we get Inf_{G/Φ(G)}^{G}(u_{1}^{2}u_{23} + u_{2}^{2}u_{13} + u_{3}^{2}u_{12}) = 0. This completes the proof. □

5. Proof of Theorem 1.2

We first consider abelian 2-groups. In the previous section we showed that if G is isomorphic to ℤ/8 or (ℤ/4)^{2} × ℤ/2, then the Euler class of the augmentation module I_{G} is zero. This implies that ζ_{G} is zero if G has a subquotient isomorphic to ℤ/8 or (ℤ/4)^{2} ×ℤ/2. But, the only abelian 2-groups that do not have any such subquotients are either elementary abelian or isomorphic to ℤ/4 × ℤ/4 or (ℤ/2)^{n} × ℤ/4 for some n. This proves one direction of Theorem 1.2 for 2-groups. For the other direction, we need to show that ζ_{G} is nonzero for these groups. We start with the calculation of ζ_{G} for G = ℤ/4 × ℤ/4.

Proposition 5.1. Let G = ℤ/4 × ℤ/4. Then, the Euler class ζ_{G} is nonzero.

Proof. Let G = ⟨a, b⟩, and let μ_{1}, μ_{2} be the generators of H^{2}(G, ℤ) dual to a and b respectively, and let μ_{12} be a generator of H^{3}(G, ℤ). We have I_{G} ≅ Inf_{G/Φ(G)}^{G}I_{G/Φ(G)} ⊕ W where W is the direct sum of all irreducible two dimensional real representations of G. Since G/Φ(G) is an elementary abelian group of order 4, it follows from an argument similar to that used in the proof of Proposition 4.3 that ζ_{G/Φ(G)} = u_{12}. Hence we get

e(Inf_{G/Φ(G)}^{G}I_{G/Φ(G)}) = Inf_{G/Φ(G)}^{G}ζ_{G/Φ(G)} = Inf_{G/Φ(G)}^{G}u_{12} = 2μ_{12}.

Now, we calculate the Euler class of W. Every two dimensional real representation of G is the underlying real representation of a one dimensional complex representation. If θ is the real representation associated to the complex representation ρ : G → ℂ^{×}, then e(θ) = c_{1}(ρ). Note that for each two dimensional real representation, the kernel is a cyclic group of order 4. In fact, there is a one-to-one correspondence between two dimensional real representations of G and its cyclic subgroups of order 4. The cyclic subgroups of G are ⟨a⟩, ⟨ab^{2}⟩, ⟨ab⟩, ⟨ab^{3}⟩, ⟨a^{2}b⟩, and ⟨b⟩. Therefore

e(W) = μ_{2}(2μ_{1} + μ_{2})(μ_{1} + μ_{2})(μ_{1} + 3μ_{2})μ_{1}(μ_{1} + 2μ_{2}) = μ_{1}^{2}μ_{2}^{2}(μ_{1} + μ_{2})^{2}

and hence ζ_{G} = 2μ_{12}μ_{1}^{2}μ_{2}^{2}(μ_{1} + μ_{2})^{2}. It is clear from Proposition 4.2 that this class is not zero in H^{∗}(G, ℤ). □

It remains to show that the Euler class is nonzero when G is either elementary abelian or isomorphic to (ℤ/2)^{n} × ℤ/4 for some n ≥ 0. For these groups we show that the Euler class is nonzero by showing that its mod 2 reduction is nonzero. Recall that the mod 2 reduction of the Euler class of a real representation V is equal to the top Stiefel-Whitney class w_{top}(V) of V. To conclude that w_{top}(I_{G}) is nonzero, we give an explicit formula for it in terms of the generators of the cohomology ring of G. It is often more convenient to express the formula for w_{top}(I_{G}) in terms of the polynomial f where

f(a_{1}, a_{2}, …, a_{m}) = ∏_{(α_{1},...,α_{m})∈(𝔽_{2})^{m}\{0}}(α_{1}a_{1} + ⋯ + α_{m}a_{m})

for tuples (a_{1}, …, a_{m}). The formula for the top Stiefel-Whitney class of an elementary abelian 2-group appears in many places (see, for example, Turygin [18]). In this case, we have

w_{top}(I_{G}) = f(x_{1}, …, x_{n})

where {x_{1}, …, x_{n}} is a set of generators of the cohomology ring H^{∗}(BG, 𝔽_{2}). Note that this is the top Dickson invariant of the polynomial algebra 𝔽_{2}[x_{1}, …, x_{n}]. In particular, the Euler class ζ_{G} is nonzero when G is an elementary abelian 2-group.

Now, we perform a similar calculation for the group G = (ℤ/2)^{n} × ℤ/4. We should note that it is possible to conclude that the top Stiefel-Whitney class is nonzero for these groups without obtaining an explicit formula, but we believe that the formula itself might also be useful. Before stating the result, we need some further notation. Let V_{i} be the one dimensional nontrivial representation inflated from the i-th term in the product of ℤ/2’s and let W_{n+1} be the irreducible two dimensional representation inflated from the ℤ/4 term of the product. Notice that the cohomology of the group G with 𝔽_{2} coefficients is

H^{∗}(G, 𝔽_{2}) ≅ 𝔽_{2}[x_{1}, …, x_{n}, s] ⊗ ∧[t]

where w_{top}(V_{i}) = x_{i} for 1 ≤ i ≤ n, w_{top}(V_{n+1}) = t, and w_{top}(W_{n+1}) = s. With this notation the formula for w_{top}(I_{G}) can be expressed as follows:

Proposition 5.2. Let G ≅ (ℤ/2)^{n} × ℤ/4 with n ≥ 0. Then,

w_{top}(I_{G}) = f(x_{1}, …, x_{n}, t)\frac{f(x_{1}^{2}, …, x_{n}^{2}, s)}{f(x_{1}^{2}, …, x_{n}^{2})}.

In particular, w_{top}(V) is nonzero in H^{∗}(G, 𝔽_{2}).

Proof. Let V be the direct sum of all the nontrivial one dimensional real representations of G. Since each nontrivial one dimensional real representation is a tensor product of elements contained in some nonempty subset of {V_{1}, ⋯, V_{n+1}}, we have w_{top}(V) = f(x_{1}, ⋯, x_{n}, t). On the other hand, I_{G} = V ⊕ W where

W = ⨁_{(γ_{1},··· ,γ_{n})∈S_{n}} V_{1}^{γ_{1}} ⊗ ⋯ ⊗ V_{n}^{γ_{n}} ⊗ W_{n+1}.

Here S_{n} is the set of n-tuples (γ_{1}, ⋯, γ_{n}) such that γ_{i} ∈ {1, 2} for all i. By the tensor product formula for one dimensional real vector bundles, we have

w(V_{1}^{γ_{1}} ⊗ ⋯ ⊗ V_{n}^{γ_{n}}) = 1 + α_{1}x_{1} + ⋯ + α_{n}x_{n}

where α_{i} denotes the mod 2 reduction of γ_{i} for each i. Using the splitting principle, we can regard the r-th Stiefel-Whitney class of the vector bundle W_{n+1} as the r-th elementary symmetric function of indeterminates a_{1} and a_{2} so that w_{1}(W_{n+1}) = a_{1} + a_{2} = 0 and w_{2}(W_{n+1}) = a_{1}a_{2} = s. Then, we have

w(V_{1}^{γ_{1}} ⊗ ⋯ ⊗ V_{n}^{γ_{n}} ⊗ W_{n+1}) = ∏_{i=1}^{2}(1 + α_{1}x_{1} + ⋯ + α_{n}x_{n} + a_{i})
= 1 + α_{1}x_{1}^{2} + ⋯ + α_{n}x_{n}^{2} + s.

Thus, the top Stiefel-Whitney class of W is given by

w_{top}(W) = ∏_{(γ_{1},··· ,γ_{n})∈S_{n}} w_{top}(V_{1}^{γ_{1}} ⊗ ⋯ ⊗ V_{n}^{γ_{n}} ⊗ W_{n+1})

= ∏_{(α_{1},··· ,α_{n})∈(𝔽_{2})^{n}}(α_{1}x_{1}^{2} + ⋯ + α_{n}x_{n}^{2} + s) = \frac{f(x_{1}^{2}, …, x_{n}^{2}, s)}{f(x_{1}^{2}, …, x_{n}^{2})}.

The formula for w_{top}(I_{G}) follows from the identity w_{top}(I_{G}) = w_{top}(V)w_{top}(W). Note that since t^{2} = 0, we can rewrite the top Stiefel-Whitney class as

w_{top}(I_{G}) = t(f(x_{1}, …, x_{n}))^{2} \frac{f(x_{1}^{2}, …, x_{n}^{2}, s)}{f(x_{1}^{2}, …, x_{n}^{2})}.

From this it is clear that w_{top}(I_{G}) is nonzero in H^{∗}(G, 𝔽_{2}). □

This completes the proof of the Theorem 1.2 for 2-groups. Now, we consider the case p > 2. We begin with the calculations for cyclic groups. Let G = ⟨g⟩ be a cyclic group of order p^{n} with p > 2. All nontrivial representations of G are two dimensional which are the underlying real representations of one dimensional complex representations. A complete list of corresponding complex representations can be given as V_{j} : g → ω^{j} where ω is the p^{n}-th root of unity and 1 ≤ j ≤ \frac{p^{n}−1}{2}. We can take α = c_{1}(V_{1}) as the generator of H^{2}(G, ℤ) ≅ ℤ/p^{n}, then we have c_{1}(V_{j}) = jα for all j. This gives

e(I_{G}) = ∏_{j=1}^{\frac{p^{n}−1}{2}} c_{1}(V_{j}) = (\frac{p^{n} − 1}{2})! α^{\frac{p^{n}−1}{2}} .

From this we conclude the following:

Lemma 5.3. Let p be an odd prime and G be a cyclic p-group. Then the Euler class ζ_{G} is nonzero if and only if G has order p or is isomorphic to ℤ/9.

Proof. Suppose that G has order p^{n}. Then, H^{2k}(G, ℤ) ≅ ℤ/p^{n} for all k ≥ 1. It follows that e(I_{G}) = 0 if and only if

(\frac{p^{n} − 1}{2})! ≡ 0 (mod p^{n}).

It is easy to see that this equation holds for all p and n except when n = 1 or when p = 3 and n = 2. □

The above lemma implies that the Euler class ζ_{G} of an abelian p-group with p > 3 vanishes if G is not elementary abelian. For p = 3, we need to be more careful. Since ζ_{G} is not zero for ℤ/9, we need to consider the next possibility, which is ℤ/9 × ℤ/3. But, this is the special case of Lemma 3.4, so ζ_{G} = 0 in this case as well. This proves one direction of Theorem 1.2 for p > 2. For the other direction, we need to show that ζ_{G} is nonzero when G is an elementary abelian p-group with p > 2. This follows easily from the structure of cohomology of elementary abelian p-groups since the 2-dimensional classes in H^{∗}((ℤ/p)^{n}, ℤ) generate a polynomial subalgebra. So, the proof of Theorem 1.2 is complete.

6. Some consequences of Theorems 1.1 and 1.2

In this section, we state and prove some corollaries of Theorems 1.1 and 1.2. We first consider the mod p reduction of the Euler class. Since most of the group cohomology calculations are done in mod p coefficients, it makes sense to consider the mod p reduction of ζ_{G}.

Corollary 6.1. Let G be a finite group and let \overline{ζ}_{G} denote the mod p reduction of the Euler class ζ_{G}. Then, \overline{ζ}_{G} is nonzero if and only if G is an elementary abelian p-group or is isomorphic to (ℤ/2)^{n} × ℤ/4 for some n ≥ 0

Proof. We only need to consider the groups where ζ_{G} is nonzero. We have already seen that when G = ℤ/4 × ℤ/4 or G = ℤ/9, the Euler class is divisible by p. So, the mod p reduction of ζ_{G} is zero in these cases. For groups isomorphic to (ℤ/2)^{n} × ℤ/4, we have already shown that ζ_{G} is nonzero by showing that its mod 2 reduction is nonzero. For elementary abelian p-groups, the mod p-reduction map is injective. So, the mod p-reduction of ζ_{G} is also nonzero for these groups. □

For p > 2, the corollary above also follows from a theorem of Serre [17]. To see this, first observe that to prove the corollary, it is enough to consider p-groups. When G is a p-group, the quotient G/Φ(G) is an elementary abelian p-group and

inf_{G/Φ(G)}^{G}\overline{ζ}_{G/Φ(G)} = λ(∏_{x∈S} β(x))^{\frac{p−1}{2}}

where λ is a nonzero scalar, S is a set of representatives of nonzero elements on each line in H^{1}(G, 𝔽_{p}), and β(x) is the image of x under the Bockstein operator. In [17], Serre proves that the product ∏_{x∈S} β(x) is zero when G is not elementary abelian. This implies that when G is a p-group which is not elementary abelian, inf_{G/Φ(G)}^{G}\overline{ζ}_{G/Φ(G)} = 0, hence \overline{ζ}_{G} = 0. It is clear that when G is an elementary abelian p-group, \overline{ζ}_{G} is nonzero.

The p = 2 case is slightly different, since in this case inf_{G/Φ(G)}^{G}\overline{ζ}_{G/Φ(G)} is equal to the product of all one dimensional classes. Serre’s theorem in [17] only gives that the product of Bocksteins of one dimensional classes is zero, so the same argument does not work in this case. The p = 2 version of Serre’s theorem has been considered in [23] and it has been proved that the product of one dimensional classes is zero exactly when G is one of the 2-groups given in the above corollary. So, the p = 2 case of the above corollary follows from the results in [23].

In the other direction, one can obtain Serre’s theorem as a consequence of Corollary 6.1. For this one needs to reduce Serre’s theorem to extra-special p-groups and apply Corollary 6.1 together with some other facts from group cohomology. For example, one needs to use the fact that if a cohomology class is detected by a central subgroup of order p, then it is a nonzero divisor. Since there are many different proofs for Serre’s theorem, this does not really provide a new way of looking at this theorem. In fact, one can see that the proof of Theorem 1.1 has many similarities to the proof of integer coefficient version of Serre’s theorem given by Evens (see Theorem 6.4.1 in [8]).

Now, we consider the square of the Euler class ζ_{G}. Note that (ζ_{G})^{2} is the Euler class of the representation I_{G} ⊕ I_{G} which can be considered as the underlying real representation of complexification of I_{G}. Note that the complexification of I_{G} is the kernel of the augmentation map ℂG → ℂ. This shows that (ζ_{G})^{2} is nothing but the top Chern class of the augmentation module of the regular complex representation of G. So, it is interesting to find exactly when this class is zero.

Corollary 6.2. Let G be a finite group. Then, (ζ_{G})^{2} ≠ 0 if and only if G is an elementary abelian p-group or is isomorphic to (ℤ/2)^{n} × ℤ/4 for some n ≥ 0. On the other hand, the mod p reduction of (ζ_{G})^{2} is nonzero if and only if G is an elementary abelian p-group.

Proof. The proof is similar to the proof of Corollary 6.1. The cases G = ℤ/4 × ℤ/4 and G = ℤ/9 do not appear here since in these cases ζ_{G} is divisible by p and both groups have cohomology with exponent p^{2}. The only group we have to be careful about is G = (ℤ/2)^{n} × ℤ/4. Note that the calculation given in Proposition 5.2 shows that the mod 2 reduction of (ζ_{G})^{2} is nonzero. But, the same calculation repeated for the top Chern class will give us that (ζ_{G})^{2} ≠ 0. The elementary abelian case is much more easier to analyze. The square of ζ_{G} is the top Dickson invariant of the polynomial subalgebra generated by two dimensional classes in H^{∗}(G, ℤ). So, both (ζ_{G})^{2} and its mod p reduction are nonzero. □

From this we can conclude the following:

Corollary 6.3. Let G be a finite group. Then, ζ_{G} is nilpotent if and only if G is not an elementary abelian p-group.

Recall that the Euler class ζ_{G} of a group G is a special case of the Euler class of a G-set X, denoted by ζ_{X}. We defined the Euler class ζ_{X} as the Euler class of the subset complex of the G-set X. One can state a similar problem: Find all G-sets X where ζ_{X} is nonzero. It is already given in [15] that ζ_{X} is zero if X is not transitive. So, we can assume X = G/H for some subgroup H ≤ G. Note that if H is normal in G, then ζ_{G/H} is the inflation of the Euler class of the quotient group G/H. In general, we can use similar ideas to reduce the calculation of ζ_{X} to smaller subquotients and smaller G-sets. The answer depends on X and the ring structure of the integral cohomology of G. At this point, we do not know any general statements that hold for all X.

7. Filtrations of Modules

In this section, we discuss some applications of our results to filtrations of modules. Suppose k denotes a field of characteristic p and kG denotes the group algebra over k. Throughout the section, we assume all kG-modules are finitely generated. For a kG-module M, we denote the j-th Heller shift of M by Ω^{j}(M). Let L_{ζ} denote the kG-module defined as the kernel of the homomorphism Ω^{n}(k) → k representing the class ζ. In [1], it has been shown that if ζ ∈ H^{n}(G, k) is represented by the kG-module extension

E : 0 → k → M_{n−1} → ⋯ → M_{0} → k → 0,

then there is a projective kG-module P such that L_{ζ} ⊕ P has a filtration

0 = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{n} = L_{ζ} ⊕ P

with L_{i}/L_{i−1} ≅ Ω^{n−i+1}(M_{i−1}) for i = 1, …, n. When ζ = 0, the module L_{ζ} is defined as the direct sum Ω(k) ⊕ Ω^{n}(k). So, if E is an extension whose extension class is zero, then there is a filtration of the form

0 = K_{0} ⊆ K_{1} ⊆ ⋯ ⊆ K_{n} = k ⊕ Ω^{n−1}(k) ⊕ P

with K_{i}/K_{i−1} ≅ Ω^{n−i}(M_{i−1}) for i = 1, …, n where P is a projective kG-module. Tensoring this sequence with a kG-module M, we get a filtration

0 = N_{0} ⊆ N_{1} ⊆ ⋯ ⊆ N_{n} = M ⊕ Ω^{n−1}(M) ⊕ Q

with N_{i}/N_{i−1} ≅ Ω^{n−i}(M_{i−1} ⊗ M) for i = 1, …, n where Q is a projective kG-module.

If the extension E is an extension with the property that all the M_{i}’s are permutation modules with no trivial summands, then we say E is an extension of proper permutation modules. Split extensions of proper permutation modules give rise to induction theorems. Using this method Carlson [4] proved that any kG-module M is a direct summand of a module that is filtered by modules induced from elementary abelian p-subgroups. He proved this result using specific split extensions of proper permutation modules where the permutation modules are induced from maximal subgroups. Such extensions exist when G is not elementary abelian as a consequence of Serre’s theorem.

Now, we will illustrate how the chain complex for the subset complex can be used to obtain similar filtrations. Note that the chain complex of the subset complex of G with coefficients in k gives an extension

0 → k → C_{n−1}(Δ(G)) ⊗_{ℤ} k → ⋯ → C_{0}(Δ(G)) ⊗_{ℤ} k → k → 0

whose extension class is the image of ζ_{G} under the map H^{n}(G, ℤ) → H^{n}(G, k) induced by tensor product with k. So, if G is a p-group which is not one of the groups listed in Theorem 1.2, then for each kG-module M, there is a projective kG-module P and a filtration

0 = N_{0} ⊆ N_{1} ⊆ ⋯ ⊆ N_{n} = M ⊕ Ω^{n−1}(M) ⊕ P

such that L_{i}/L_{i−1} ≅ Ω^{n−i}(C_{i−1}(Δ(G))⊗_{ℤ}M) for i = 1, …, n. Instead of the chain complex, we could have used the cochain complex of the subset complex. In that case, we get the following:

Corollary 7.1. Let G be a finite group and let k be a field of characteristic p. If G is not one of the groups listed in Theorem 1.2, then for every kG-module M, there is a projective kG-module P and a filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M ⊕ Ω^{n−1}(M) ⊕ P

such that M_{i}/M_{i−1} ≅ Ω^{n−i}(C^{n−i}(Δ(G)) ⊗_{ℤ} M) for i = 1, …, n.

In [15], an explicit formula for permutation modules appearing in C^{j}(Δ(X)) is given. So, using this formula we can give upper bounds for the dimension of a kG-module M in terms of the Heller shifts of the projective free parts of the restrictions of M to its proper subgroups. This approach was used to obtain upper bounds for essential endotrivial modules of extraspecial p-groups in [5]. For small groups our bounds are not as good as the bounds obtained in [5], but we expect that for groups with large cohomology lengths, this method will give better bounds than the ones given in [5]. Unfortunately, we were not able to verify this.

8. Borsuk-Ulam type theorems

In this section, we give some applications of our results to the existence of Borsuk-Ulam type theorems. Recall that the Borsuk-Ulam theorem says that every continuous map f : S^{n} → ℝ^{m} maps some pair of antipodal points on S^{n} to the same point in ℝ^{m} provided that m ≤ n. To generalize this theorem, one defines the notion of coincidence set. Suppose M is a topological manifold with a free G-action and let f : M → ℝ^{m} be a continuous map. The coincidence set of f is defined as the set

A(f) = {x ∈ M | f(x) = f(gx) for all g ∈ G}.

We say that M has a Borsuk-Ulam type theorem if there is an appropriate condition on m such that the coincidence set is nonempty. The Borsuk-Ulam theorem corresponds to the case where M = S^{n} and G = ℤ/2 acting with antipodal action on M. Another example is given recently by Turygin [18] where M is a product of (mod p) homology spheres and G is an elementary abelian p-group acting on M by a product action. Some other generalizations of these types of theorems are discussed in [18]. It is interesting to note that one of these generalizations is related to a well known theorem of Milnor [12] which states that if a finite group G acts freely on a sphere, then every element of order 2 in G must be central.

Now we will discuss a topological condition for the existence of a Borsuk-Ulam type theorem for a manifold M with a free G action. Consider the real vector bundle

ξ : M ×_{G} I_{G}^{⊕m} → M/G

with classifying map q : M/G → BG where I_{G} denotes the augmentation module. It is easy to see that there exists a continuous map f : M → ℝ^{m} with empty coincidence set if and only if ξ has a nonzero section (see Lemma 1.1 in [18]). So, to prove a Borsuk-Ulam type theorem for M, it is enough to show that, under suitable conditions on m, the bundle ξ has a nontrivial Euler class. The Euler class of this bundle is the image of the Euler class of the universal bundle ξ̂: EG ×_{G} I_{G}^{⊕m} → BG under the map q^{∗} : H^{∗}(BG, ℤ) → H^{∗}(M/G, ℤ). Note that the Euler class of ξ̂ is the same as the m-th power of ζ_{G}. We can summarize these in the following way:

Proposition 8.1 (Sarkaria [16], Turygin[18]). Let M be a topological manifold with a free action of a finite group G. Suppose that (q^{∗}(ζ_{G}))^{m} ≠ 0 where q^{∗} : H^{∗}(BG, ℤ) → H^{∗}(M/G, ℤ) is the map induced by the classifying map q : M/G → BG. Then, every continuous map f : M → ℝ^{m} has a nonempty coincidence set.

This shows that we can find some Borsuk-Ulam type theorems for the groups listed in Theorem 1.2 if we choose the manifold M in an appropriate way. This is how Turygin [18] obtains the Borsuk-Ulam type theorem for products of spheres with free elementary abelian group action. However, it is clear that when ζ_{G} = 0, we cannot find any G-free space M satisfying the condition in Proposition 8.1. So, we conclude the following.

Corollary 8.2. If G is not one of the groups listed in Theorem 1.2, then one cannot obtain Borsuk-Ulam type theorems for any G-free topological manifold using the cohomological Euler class.

This does not say that there are no Borsuk-Ulam theorems since it is still possible to use some other cohomology theory and conclude that the bundle ξ : M ×_{G} I_{G}^{⊕m} → M/G has no nonzero section.

Now, we explain another application of our results. For this we need to define the Borsuk-Ulam property. Note that another way to state the Borsuk-Ulam theorem is the following: For every antipodal preserving map f : S^{n} → ℝ^{n}, there exists an x ∈ S^{n} such that f(x) = 0. Note that if we take G = ℤ/2, then ℝ^{n} can be thought of as the ℝG-module I_{G}^{⊕n} and S^{n} can be thought of as the (n + 1)-fold join G^{∗(n+1)} with usual G-action. One generalizes this situation as follows:

Definition 8.3. Let G be a finite group and let V be an n-dimensional real representation of G. The representation V is said to have a Borsuk-Ulam property if any continuous G-map f : G^{∗(n+1)} → V has a zero.

Borsuk-Ulam theorem is equivalent to the statement that I_{G}^{⊕n} has Borsuk-Ulam property when G = ℤ/2. One can ask if the similar statement holds for other groups. In [16], Sarkaria shows that an n-dimensional real representation V has the Borsuk-Ulam property if and only if the Euler class e(V) ∈ H^{n}(G, ℤ̃) is nonzero (see Theorem 1, [16]). As a consequence of Theorems 1.1 and 1.2, we obtain the following:

Corollary 8.4. Let G be a finite group. The augmentation module I_{G} has Borsuk-Ulam property if and only if G is one of the groups listed in Theorem 1.2.

Borsuk-Ulam property is closely related to the continuous version of Tverberg Theorem in combinatorics. The Tverberg Theorem can be stated as follows:

Theorem 8.5 (Tverberg). For any affine map f from the standard (q − 1)(d + 1)-simplex to an affine d-space, there are q disjoint faces σ_{1}, ⋯, σ_{q} such that the intersection f(σ_{1}) ∩ ⋯ ∩ f(σ_{q}) is nonempty.

The continuous Tverberg theorem is a generalization of this theorem to continuous maps. It is known that the continuous Tverberg theorem holds for all prime powers q = p^{k}. There is a very nice paper by Sarkaria [16] where the connection between the continuous Tverberg problem and the Borsuk-Ulam property has been described. In this article, Sarkaria shows that the continuous Tverberg theorem holds for some q and d, if there exists a group G of order q for which I_{G}^{⊕(d+1)} has the Borsuk-Ulam property. There is a theorem by Özaydın [13], proved independently by Volovikov [20], which states the following:

Proposition 8.6 ([16], Theorem 4). Every representation V of an elementary abelian group which does not contain the trivial representation has the Borsuk-Ulam property.

Özaydın-Volovikov theorem solves the topological Tverberg problem when q is a prime power. Note that Corollary 6.3 not only implies the Özaydın-Volovikov theorem, but also shows that elementary abelian p-groups are the only groups where the Özaydın-Volovikov theorem holds. It is still an open problem to show if the Tverberg theorem holds when q is a composite number. For more detailed discussion about this problem, we refer the reader to [10] and [16].

9. Hypercohomology spectral sequences

In this section, we consider the hypercohomology spectral sequences associated to the subset complex. Given a cochain complex C^{∗} of ℤG-modules, we can form a double complex Hom_{ℤG}(P_{∗}, C^{∗}) where P_{∗} is a projective resolution of ℤ over ℤG. The cohomology of this cochain complex is called the hypercohomology of G with coefficients in C^{∗} and usually denoted by H^{∗}(G, C^{∗}). Since this is the cohomology of a double complex, there are two spectral sequences which converge to H^{∗}(G, C^{∗}). The first spectral sequence, which is usually called the cohomology spectral sequence, has E_{2}-page

E_{2}^{p,q} = H^{p}(G, H^{q}(C^{∗}, ℤ)) ⇒ H^{p+q}(G, C^{∗}).

We have a second spectral sequence with E_{1}-page

E_{1}^{p,q} = H^{q}(G, C^{p}) ⇒ H^{p+q}(G, C^{∗})

which is called the isotropy spectral sequence. We refer the reader to the book by Brown [6] for more information on hypercohomology spectral sequences.

In [15], Reiner and Webb considers the spectral sequences which are isomorphic to the hypercohomology spectral sequences for C^{∗}(Δ(G)). Since Δ(G) is contractible, the cohomology spectral sequence gives

H^{∗}(G, C^{∗}(Δ(G))) ≅ H^{∗}(G, ℤ).

Reiner and Webb study the isotropy spectral sequence for this hypercohomology group to obtain information about integral cohomology groups of G. For this, they obtain a formula which gives the (signed) permutation modules appearing in C_{n}(Δ(G)) for each n. The formula is in the form of a power series with coefficients in the Burnside ring B(G) of G. Once this power series is calculated, it becomes possible to calculate the E_{1} page of the isotropy spectral sequence. Reiner and Webb illustrate how this method works by giving estimates for the orders of some integral cohomology groups of the Dihedral group of order 8. Theorems 1.1 and 1.2 have some immediate applications to Reiner and Webb’s hypercohomology calculations. For example, by Corollary 7.7 in [15], we obtain the following:

Corollary 9.1. Let G be a finite group which is not one of the groups listed in Theorem 1.2. Then, E_{∞}^{r,|G|−1} = 0 for all r in the isotropy spectral sequence for the subset complex Δ(G).

Another hypercohomology calculation that is interesting to consider is the hypercohomology of G with coefficients in the cochain complex of the boundary ∂Δ(G) of the subset complex. Since ∂Δ(G) is homeomorphic to a sphere, the E_{2}-page of the cohomology spectral sequence is a two line spectral sequence. If |G| = n + 1, then E_{2}^{p,q} is nonzero only when q = 0 or q = n − 1. Moreover, the differential d_{n} is given by the multiplication with ζ_{G} ∈ H^{n}(G, ℤ̃). So, if G is not one of the groups listed in Theorem 1.2, then the cohomology spectral sequence for ∂Δ(G) collapses at E_{2}-page, and we have

H^{i}(G, C^{∗}(∂Δ(G))) ≅ \begin{cases}H^{i}(G, ℤ) if i < n − 1 \\ H^{i}(G, ℤ) ⊕ H^{i−n+1}(G, ℤ) if i ≥ n − 1. \end{cases}

The collapsing of this spectral sequence is important for theoretical reasons as well as computational reasons. Note that when the spectral sequence collapses at the E_{2}-page, the edge homomorphism

π^{∗} : H^{∗}(G, ℤ) → H^{∗}(G, C_{∗}(∂Δ(G)))

is injective. The hypercohomology of the chain complex C_{∗}(∂Δ(G)) is isomorphic to the equivariant cohomology of the G-complex ∂Δ(G). It is known that the equivariant cohomology of a finite G-CW -complex X has an injective edge homomorphism if X has a fixed point. When G is an elementary abelian p-group, the converse is also known to hold in mod p coefficients as a consequence of the well-known Localization Theorem. The boundary of the subset complex has no fixed points but its edge homomorphism is zero when G is not one of the groups listed in Theorem 1.2. Therefore, ∂Δ(G) can be considered as a counterexample to the Localization Theorem for arbitrary groups. Other examples of such spaces are also known, for example, the projectivization of an irreducible complex representation has zero edge homomorphism but has no fixed points when the representation is of dimension greater than 2.

In [14], Pakianathan and Yalçın show that if there exists a topological G-space Y which has no G-fixed point and for which the edge homomorphism π^{∗} : H^{∗}(G) → H_{G}^{∗}(Y) is injective, then the essential ideal Ess^{∗}(G) is nilpotent with nilpotency degree less than or equal to the dimension of Y. Recall that the essential ideal of the group G is defined as

Ess^{∗}(G) = Ker{∏ res_{H}^{G} : H^{∗}(G) → ∏_{H<G} H^{∗}(H)}.

Since ∂Δ(G) has no fixed points, our main theorem also implies the nilpotency of essential ideal for nonabelian groups. The nilpotency of essential ideal has many consequences in group cohomology such as Quillen’s F-injectivity theorem and Serre’s theorem.

Although the hypercohomology spectral sequence for the subset complex is a useful tool to obtain information about the cohomology of the group, we believe that it would be very difficult to calculate group cohomology using this method. One of the difficulties is that we do not know at which page the isotropy spectral sequence converges. It is an interesting problem to find group theoretical conditions which make the isotropy spectral sequence converge at a certain page.

Another problem with the isotropy spectral sequence of ∂Δ(G) is that the subset complex is not an admissible simplicial complex, so the product structure on the spectral sequence is not compatible with the product structure of the integral cohomology ring. That means that we can only calculate cohomology groups of G, but not the cohomology ring of G. The following is an example of a subset complex where the product structure of the isotropy spectral sequence is not compatible with the product structure of the group cohomology.

Example 9.2. Let G = ⟨a, b⟩ be an elementary abelian group of order 4. Then the corresponding complex ∂Δ(G) can be pictured as a tetrahedron and the Euler class of Δ(G) is ζ_{G} = μ_{12} where μ_{12} is a generator of H^{3}(G, ℤ). Recall that in this case the integral cohomology ring of G has the structure

H^{∗}(G, ℤ) ≅ ℤ/2[μ_{1}, μ_{2}, μ_{12}]/⟨μ_{1}μ_{2}(μ_{1} + μ_{2}) − μ_{12}^{2}⟩.

The differential d_{3} is given by multiplication with μ_{12} which is a nonzero divisor. This gives that H^{n}(G, C_{∗}) ≅ H^{n}(G, ℤ)/⟨μ_{12}⟩ for n ≠ 0 and H^{2}(G, C^{∗}) ≅ H^{2}(G, ℤ) ⊕ ℤ. On the other hand, the chain complex of ∂Δ(G) is

0 → ℤ[G] → ⊕_{i=1}^{3}ℤ̃[G/H_{i}] → ℤ[G] → 0

where H_{1}, H_{2}, H_{3} are the cyclic subgroups of order 2 in G. So, the E_{1}-page of the isotropy spectral sequence is as follows:

ℤ ℤ •  ×  • ⊕_{3}ℤ/2 • •⊕_{3}ℤ/2 • ×⊕_{3}ℤ/2 ×  The E_{1} − 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It is easy to see that all differentials d_{i} are zero for i ≥ 1 and hence E_{∞} = E_{1}. This gives

H^{∗}(G, ℤ) ≅ ⨁_{i=1}^{3} H^{∗}(H_{i}, ℤ̃)

except at the dimensions 0 and 2. Note that the product E_{1}^{1,1} ⊗ E_{1}^{1,1} → E_{1}^{2,2} is zero, so all the 2-dimensional classes are nilpotent with respect to the product in the spectral sequence, but this is not true in the integral cohomology of the group. This shows that the product structure of spectral sequence is not compatible with the product structure in the integral cohomology of the group.

It is known that when a simplicial complex has an admissible G-action, the cohomology structure is compatible with the product structure of the associated isotropy spectral sequence (see [6]). Recall that a G-action on a simplicial complex is called admissible if it satisfies the condition that if a simplex is fixed by an element in G then its vertices are also fixed. The induced action on barycentric subdivision of a simplicial complex is an admissible action. Therefore to avoid problems arising with the product structure, one should consider the barycentric subdivision of the complex ∂Δ(G).

10. Barycentric subdivision of the subset complex

In [15], Reiner and Webb give a formula which provides a complete description of the (signed) permutation modules involved in the chain complex of Δ(G). Using this formula, one calculates the E_{1}-page of the isotropy spectral sequence for ∂Δ(G). To be able to do similar calculations with the barycentric subdivision of ∂Δ(G), we need to find a similar formula for the barycentric subdivision of ∂Δ(G). We first describe the formula given by Reiner and Webb and show how a similar formula for the barycentric subdivision of ∂Δ(G) can be obtained.

A graded G-set is a G-set Ω partitioned as Ω = Ω(0) ∪ Ω(1) ∪ Ω(2) ∪ ⋯ where each component Ω(i) is a G-set. Given a graded G-set, one can form a Poincaré series

P_{Ω}(t) = ∑_{i≥0}[Ω(i)]t^{i}

where [Ω(i)] denotes the isomorphism class of the set Ω(i). We can consider this series as a power series with coefficients in the Burnside ring B(G). By using the properties of the Burnside ring and the Gluck idempotent formula, Webb [15] gives a formula for this Poincaré series as

(1)  P_{Ω}(t) = ∑_{K≤H≤G} \frac{[G/K]μ(K, H)f_{H}(t)}{|G : K|}

where f_{H}(t) = ∑_{i≥0} |Ω(i)^{H}|t^{i} and μ denotes the Möbius function on the poset of subgroups of G. Note that in the above formula [G/K] denotes the isomorphism class of the transitive G-set with isotropy subgroup K.

For a G-simplicial complex Γ, the Poincaré series P_{Γ}(t) associated to Γ is defined as the Poincare series of the graded G-set

Γ = Γ(0) ∪ Γ(1) ∪ Γ(2) ∪ ⋯

where Γ(0) = [G/G] and Γ(i) is the isomorphism class of the set of (i − 1)-simplices of Γ for i ≥ 1. If we apply this definition to the subset complex Δ(X), we get a Poincaré series of the form

P_{Δ(X)}(t) = Λ^{0}(X) + Λ^{1}(X)t + ⋯ + Λ^{n+1}(X)t^{n+1}

where |X| = n + 1 and Λ^{i}(X) is the i-th exterior power of X. In [21], Webb simplifies the formula in Equation 1 for this case and obtains the following:

Proposition 10.1 (Proposition 1.4, [21]). Let X be a finite G-set. Then

P_{Δ(X)}(t) = ∑_{K≤H≤G} \frac{[G/K]μ(K, H)f_{H}(t)}{|G : K|}

where f_{H}(t) = ∏_{i=1}^{m}(1 + t^{|X_{i}|}) when X = X_{1} ∪ ⋯ ∪ X_{m} is the decomposition of X into H orbits.

For a G-set X, let Bary(X) denote the barycentric subdivision of ∂Δ(X). We are interested in finding a similar formula for the Poincaré series of Bary(X). Let us call Bary(X)(i) the i-th T -power of the set X and denote it with T^{i}(X). With this notation we have

P_{Bary(X)}(t) = T^{0}(X) + T^{1}(X)t + ⋯ + T^{n}(X)t^{n}

where |X| = n + 1. We have the following alternative description for T -powers of a G-set.

Lemma 10.2. For a G-set X, the i-th T -power T^{i}(X) is isomorphic to the set of surjective maps f : X → {1, 2, …, i + 1}.

Proof. By definition, the i-th T -power of X is the set of proper chains

∅ ⊂ A_{1} ⊂ A_{2} ⊂ ⋯ ⊂ A_{i} ⊂ X.

It is easy to see that this set is isomorphic to the set of sequences B_{1}, B_{2}, …, B_{i+1} of nonempty disjoint sets whose union is X. But, this is isomorphic to the set of surjections f : X → {1, 2, …, i + 1}. □

Let us denote the size of the set T^{i}(\{1, 2, …, m\}) by [{m \atop i}]. This is analogous to the similar notation |Λ^{i}(\{1, 2, …, m\})| = \binom{m}{i} for exterior powers. Note that

[{m \atop i}] = (i + 1)! S(m, i + 1)

where S(m, i + 1) denotes the Stirling number of second type which is the number of ways to partition the set {1, ⋯, m} into i + 1 disjoint subsets. The Poincaré series for Bary(X) is given by the following formula.

Proposition 10.3. Let X be a G-set and let Bary(X) denote the barycentric subdivision of the boundary of the subset complex Δ(X). Then,

P_{Bary(X)}(t) = ∑_{K≤H≤G} \frac{[G/K]μ(K, H)F_{|X/H|}(t)}{|G : K|}

where F_{m}(t) = [{m \atop 0}] + [{m \atop 1}]t + ⋯ + [{m \atop m−1}]t^{m−1} for m ≥ 1.

Proof. By Equation 1, it is enough to show that |T^{i}(X)^{H}| = [{|X/H| \atop i}]. Note that by Lemma 10.2, the fixed point set T^{i}(X)^{H} is in one-to-one correspondence with surjective maps f : X/H → {1, 2, …, i + 1}. So, the result follows. □

The polynomials F_{m}(t) can be easily calculated using the induction formula

[{m \atop i}] = ([{m − 1 \atop i}] + [{m − 1 \atop i − 1}])(i + 1).

The first six terms of the series F_{m}(t) are as follows:

F_{1}(t) = 1

F_{2}(t) = 1 + 2t
F_{3}(t) = 1 + 6t + 6t^{2}
F_{4}(t) = 1 + 14t + 36t^{2} + 24t^{3}
F_{5}(t) = 1 + 30t + 150t^{2} + 240t^{3} + 120t^{4}
F_{6}(t) = 1 + 62t + 540t^{2} + 1560t^{3} + 1800t^{4} + 720t^{5}

We conclude this section with an example which illustrates how one can obtain information about the product structure using the complex Bary(X).

Example 10.4. Let G be an elementary abelian group of order 4, and let H_{1}, H_{2}, H_{3} denote its subgroups of order 2. Consider Bary(G), the barycentric subdivision of ∂Δ(G). The cohomology spectral sequence for this complex is the same as the one given in Example 9.2. Let us look at the isotropy spectral sequence closely. By Proposition 10.3, the coefficient of [G/H_{i}] in P_{Bary(X)}(t) is

\frac{1}{2}[F_{2}(t) − F_{1}(t)] = \frac{1}{2}[(1 + 2t) − 1] = t

for each i = 1, 2, 3 and the coefficient of [G/1] is

\frac{1}{4}[F_{4}(t) − 3F_{2}(t) + 2F_{1}(t)] = 2t + 9t^{2} + 6t^{3}.

This means that the chain complex for Bary(X) is given by

0 → ⊕_{6}ℤ[G/1] → ⊕_{9}ℤ[G/1] → ⊕_{i}ℤ[G/H_{i}] ⊕ (⊕_{2}ℤ[G/1]) → 0.

The differential d_{1} on the E_{1}-page is nonzero only on the bottom line and can easily be calculated. We obtain that the E_{2}-page looks like the following:

ℤ •  • × ×⊕_{3}ℤ/2 • •⊕_{3}ℤ/2  The E_{2} − page 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Since there are no further differentials, the spectral sequence collapses at the E_{2}-page. Note that if u_{1} and u_{2} are generators of H^{2}(G, ℤ) corresponding to projections on each factor, then they map to (1, 1, 0) and (0, 1, 1) in E_{3}^{0,2}. So, we obtain the relation u_{1}u_{2}(u_{1} + u_{2}) = 0 as a consequence of the product structure on E_{3}. This shows that the product structure of the spectral sequence is compatible with the product structure of the group cohomology.

References

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Asli Güçlükan, Department of Mathematics, Bilkent University, Bilkent, Ankara, Turkey. Current Address: University of Rochester, Department of Mathematics, Rochester, NY, USA

E-mail address: guclukan@fen.bilkent.edu.tr

Ergün Yalçın, Department of Mathematics, Bilkent University, Bilkent, Ankara, Turkey. Current Address: McMaster University, Department of Mathematics and Statistics, Hamilton, ON, Canada

E-mail address: yalcine@fen.bilkent.edu.tr