ACYCLIC CHAIN COMPLEXES OVER THE ORBIT CATEGORY

IAN HAMBLETON AND ERGÜN YALÇIN

Abstract. Chain complexes of finitely generated free modules over orbit categories provide natural algebraic models for finite G-CW-complexes with prescribed isotropy. We prove a p-hypoelementary Dress induction theorem for K-theory over the orbit category, and use it to re-interpret some results of Oliver and Kropholler-Wall on acyclic complexes.

1. Introduction

A good algebraic setting for studying actions of a group G with isotropy in a given family of subgroups ℱ is provided by the category of R-modules over the orbit category Γ = Or_{ℱ} G, where R is a commutative ring with unit. This theory was established by Bredon [5], tom Dieck [10] and Lück [20], and further developed by many authors (see, for example, Jackowski-McClure-Oliver [18, §5], Brady-Leary-Nucinkis [4], Symonds [24], [25]).

The category of RΓ -modules is an abelian category with Hom and tensor product, and has enough projectives for standard homological algebra. In this paper, we will use projective chain complexes over the orbit category of a finite group to study acyclic G-CW complexes. In Section 2 we give an orbit category version of an induction result of Dress [12]. In Sections 3 and 4 we re-interpret some results of Oliver [21] and Kropholler-Wall [19] in terms of algebra over the orbit category.

2. Dress induction over the orbit category

Let G be a finite group and let R = ℤ̂_{p} or R = ℤ/p, for some prime p. We note that the Krull-Schmidt theorem holds for finitely-generated RG-modules. Let A(RG) denote the Grothendieck ring of isomorphism classes of R-torsion free RG-modules, with addition given by direct sums and product given by tensor product ⊗_{R}. By the Krull-Schmidt theorem, A(RG) is ℤ-torsion free.

Andreas Dress [12, Theorem 7] proved that A(RG) is rationally generated by induction from all the p-hypoelementary subgroups of G, and detected by restriction to the same collection of subgroups (see also Bouc [3, Cor. 3.5.8] for an exposition). Recall that a subgroup H ≤ G is called p-hypoelementary if it has a normal p-subgroup P ⊴ H such that H/P is cyclic of order prime to p. We denote the class of p-hypoelementary group by 𝒢_{p}^{1}. In this section we will give a version of the result of Dress for modules over the orbit category.

Date: Sept. 3, 2009.

Research partially supported by NSERC Discovery Grant A4000. The first author would like to thank the Max Planck Institut für Mathematik in Bonn for its hospitality and support while part of this work was done. The second author is partially supported by TÜBA-GEBİP/2005-16.

Let Γ denote the orbit category of G with respect to the family ℱ of p-subgroups in G. Free RΓ -modules are direct sums of the modules R[G/Q^{?} ], for Q ∈ ℱ, where

R[G/Q^{?} ](G/V) = R Mor_{G}(G/V, G/Q),

and the Grothendieck group of finitely-generated projective RΓ -modules is denoted K_{0}(RΓ) (see [20, §10] for the definition and properties of this K_{0} functor). We remark that K_{0}(RΓ) is a Mackey functor under the natural operations of induction Ind_{H}^{G} and restriction Res_{H}^{G}, with respect to subgroups H ≤ G.

Let RE_{G} denote the exact category of finitely-generated R-torsion free RΓ -modules, of finite projective length over RΓ, with exactness structure given by the short exact sequences of RΓ -modules.

Example 2.1. Every RΓ -module of the form R[G/H^{?} ], H ≤ G, admits a finite length projective resolution. This follows from the orbit category version of Rim’s theorem (see [15, Theorem 3.8]). However, the adjunction formula [20, 17.21] shows that, for example, the module E_{1}(R) = I_{1}(R) does not have a finite length projective resolution if G = ℤ/p.

We note that K_{0}(RE_{G}) is a ring under the operations of direct sum and tensor product ⊗_{R}, with unit \underline{R} = R[G/G^{?} ] the constant RΓ -module. Moreover, K_{0}(RE_{G}) also has the structure of a Mackey functor with respect to Ind_{H}^{G} and Res_{H}^{G}, and hence is a Green ring (via the product formulas of [20, 10.26], and the observation that the diagonal functor Δ: Γ → Γ × Γ is admissible [20, p. 203]). The natural map K_{0}(RΓ_{G}) → K_{0}(RE_{G}), sending [P] ↦ [P], is called the Cartan map.

Lemma 2.2 (Grothendieck, Swan). The Cartan map K_{0}(RΓ_{G}) ⟶^{≈} K_{0}(RE_{G}) is an isomorphism of Mackey functors.

Proof. If M is an RΓ -module and P_{∗} → M is a projective resolution, we may define χ: K_{0}(E_{G}) → K_{0}(RΓ_{G}) by

χ(M) = ∑(−1)^{i}[P_{i}] ∈ K_{0}(RΓ).

The Cartan map K_{0}(RΓ_{G}) → K_{0}(RE_{G}) is compatible with induction and restriction, and χ gives an inverse map as in Swan [23, Thm. 1.1], or Curtis-Reiner [8, 38.50]. □

Lemma 2.3. K_{0}(RΓ_{G}) and K̃_{0}(RΓ_{G}) are ℤ-torsion-free (for the orbit category with respect to any family ℱ of subgroups).

Proof. There is a (split) short exact sequence (see [20, 10.42])

0 → K_{0}^{f}(RΓ_{G}) → K_{0}(RΓ_{G}) → K̃_{0}(RΓ_{G}) → 0

where K_{0}^{f}(RΓ) denotes K_{0} of the exact category of finitely-generated free RΓ -modules. In addition, there is a natural isomorphism (see Lück [20, 10.34]):

K_{0}(RΓ_{G}) ≅ ⨁_{[Q]∈Iso(Γ_{G})} K_{0}(R[N_{G}(Q)/Q])

induced by the inverse functors S = (S_{Q}) and E = (E_{Q}). Here Iso(Γ_{G}) denotes the isomorphism classes of objects in Γ_{G}, or equivalently the G-conjugacy classes of subgroups Q ∈ ℱ. By the Krull-Schmidt theorem, all of the groups K_{0}(R[N_{G}(Q)/Q]) are ℤ-torsion free. □

Here is the main result of this section.

Theorem 2.4. Let Γ denote the orbit category of a finite group G with respect to the family of p-subgroups, for some prime p, and let R = ℤ̂_{p} or R = ℤ/p. Then K_{0}(RΓ_{G})⊗ℚ and K̃_{0}(RΓ_{G}) ⊗ ℚ are computable from the p-hypoelementary subgroups of G.

Since K̃_{0}(RΓ_{G}) is ℤ-torsion free, we have the immediate consequence:

Corollary 2.5. K_{0}^{f}(RΓ_{G}), K_{0}(RΓ_{G}) and K̃_{0}(RΓ_{G}) are detected by restriction to the sum of K̃_{0}(RΓ_{H}), for all H ∈ 𝒢_{p}^{1}.

For p and q primes, let 𝒢_{p}^{q} denote the class of finite groups which have a normal subgroup H ∈ 𝒢_{p}^{1}, with q-power order quotient group. Let 𝒢_{p} = ⋃_{q} 𝒢_{p}^{q} (see Dress [12, §9] and Oliver [21]).

Corollary 2.6. K_{0}^{f}(RΓ_{G}), K_{0}(RΓ_{G}) and K̃_{0}(RΓ_{G}) are computable by induction or restriction from the family of subgroups in 𝒢_{p}.

Proof. Note that 𝒢_{p}^{q} = hyper_{q}-𝒢_{p}^{1} in the terminology of Dress, so the result follows from Corollary 2.5 and Dress induction [12, p. 207], [17, 3.3]. □

The proof of Theorem 2.4. The Burnside quotient Green ring A_{K} of K_{0}(RE_{G}) is isomorphic to the subring generated by the modules R[G/H^{?} ], for all H ≤ G (see [17, Remark 2.4]). By Lemma 2.2 it follows that A_{K} is also the Burnside quotient Green ring of the Mackey functor K_{0}(RΓ_{G}). Since K̃_{0}(RΓ_{G}) is a quotient Mackey functor of K_{0}(RΓ_{G}), it is also a Green module over A_{K} (see [17, §2D]). By Dress induction [13], it suffices to show that A_{K} ⊗ ℚ is generated by induction from the family of p-hypoelementary subgroups of G.

For each subgroup H ≤ G, there is a covariant functor F: Γ_{H} → Γ_{G} of orbit categories with respect to ℱ, such that Ind_{F} = Ind_{H}^{G} and Res_{F} = Res_{H}^{G} on K-theory. By [20, 10.34] there is a commutative diagram

(2.7)  \xymatrix{K_{0}(RΓ_{H}) \ar[r]^{^{Ind^{G}^{H}}} & K_{0}(RΓ_{G}) \ar[d]^{S(RΓ_{G})}_{≈} \\ ⨁_{[V]∈Iso(Γ_{H})} K_{0}(R[N_{H}(V)/V]) \ar[r]^{F_{∗}} \ar[u]^{E(RΓ_{H})}_{≈} & ⨁_{[Q]∈Iso(Γ_{G})} K_{0}(R[N_{G}(Q)/Q])}

where the vertical maps are the splitting isomorphisms, and the lower horizontal map F_{∗} is the sum of the induction maps corresponding to the subgroups N_{H}(V) ⊂ N_{G}(V), for V ≤ H, and V ∈ ℱ (see [20, 10.12]). There is a similar diagram for Res_{H}^{G} and the contravariant map F^{∗} using [20, 10.15], but the formula for F^{∗} is more complicated. The functors E and S are inverse pairs of natural equivalences, and we have the formulas

F_{∗} = S(RΓ_{G}) ∘ Ind_{H}^{G} ∘E(RΓ_{H})

and

F^{∗} = S(RΓ_{H}) ∘ Res_{H}^{G} ∘E(RΓ_{G})

for the induced maps in diagram (2.7). The component of F_{∗} at [Q] ∈ Iso(Γ_{G}) will be denoted p_{Q}F_{∗}, and similarly p_{V} F^{∗} will denote the component of F^{∗} at [V] ∈ Iso(Γ_{H}).

We wish to show that there exist rational numbers {r_{H} | H ∈ 𝒢_{p}^{1}} such that

(2.8)  a = ∑_{H∈𝒢_{p}^{1}} r_{H} Ind_{H}^{G}(Res_{H}^{G}(a))

for any a ∈ K_{0}(RΓ_{G}). This is equivalent to the statement that A_{K} ⊗ ℚ is generated by induction from the family of p-hypoelementary subgroups of G.

We will establish formula (2.8) by induction on the support

supp(a) := {[K] ∈ Iso(Γ_{G}) | S_{K}(a) ≠ 0}

of an element a ∈ K_{0}(R[N_{G}(Q)/Q]), where the support sets are partially ordered by conjugation-inclusion. Since E(RΓ_{G}) is an isomorphism, we may assume that a = E_{Q}(a_{Q}), for some a_{Q} ∈ K_{0}(R[N_{G}(Q)/Q]). Let us also assume that formula (2.8) holds for all elements b with supp(b) < supp(a) = {K ≤_{G} Q}.

From the expressions above for F_{∗} and F^{∗} we have the relation

(2.9)  S(RΓ_{G}) (Ind_{H}^{G}(Res_{H}^{G}(a))) = F_{∗}(F^{∗}(S(RΓ_{G})(a))) = F_{∗}(F^{∗}(a_{Q}))

so we need to compute p_{K}F_{∗}(F^{∗}(a_{Q})), for all subgroups K ∈ ℱ.

First, we compute p_{Q}F_{∗}(F^{∗}(a_{Q})), for a_{Q} ∈ K_{0}(R[N_{G}(Q)/Q]). By [20, 10.12], the only non-zero components of p_{Q}F_{∗} are given by the images of

Ind_{W_{H}(K)}^{W_{G}(Q)} = (S_{Q} ∘ Ind_{F} ∘E_{K})_{∗}

corresponding to the objects H/K in Γ_{H} such that G/Q = F (H/K) in Γ_{G}. In other words, we need only the subgroups K ≤ H such that K is a G-conjugate of Q. Without loss of generality, we may assume that Q ≤ H since p_{Q}F_{∗}(F^{∗}(a_{Q})) = 0 unless Q is conjugate to a subgroup of H.

Therefore, we only need to consider the components p_{K}F^{∗} of F^{∗} which have the form

M ↦ M ⊗_{R[W_{G}(Q)]} R Hom_{G}(F (H/K), G/Q)

where G/Q = F (H/K) and M is a right R[W_{G}(Q)]-module, as given in [20, 10.15]. We are using the formula

R Irr(H/K, G/Q) = S_{K} (R Hom_{G}(F (?), G/Q)) = R Hom_{G}(F (H/K), G/Q) .

The right-hand side is a right R[W_{G}(K)]-module through the natural action of R[W_{H}(K)] on H/K. But since Hom_{G}(F (H/K), G/Q) = W_{G}(Q) whenever F (H/K) = G/Q, each of these components of F^{∗} is just the usual restriction Res_{W_{H}(K)}^{W_{G}(K)} composed with a conjugationinduced isomorphism W_{G}(K) ≅ W_{G}(Q).

It follows that p_{Q}F_{∗}(F^{∗}(a_{Q})) is a sum of terms indexed by the H-conjugacy classes of subgroups K ≤ H such that K is G-conjugate to Q. We have the formula

(2.10)  p_{Q}F_{∗}(F^{∗}(a_{Q})) = ∑_{{K≤H, K^{g}=Q}_{H}} Ind_{W_{H^{g}} (Q)}^{W_{G}(Q)} (Res_{W_{H^{g}} (Q)}^{W_{G}(Q)}(a_{Q})) ,

where each term in the sum is obtained by (i) choosing an H-conjugacy class representative K ≤ H, and then (ii) picking an element g ∈ G with K^{g} = Q.

Note that the individual terms on the right-hand side of formula (2.10) are independent of the choices made: K^{g_{1}} = K^{g_{2}} = Q implies that W_{H^{g_{1}}}(Q) and W_{H^{g_{2}}}(Q) are conjugate in W_{G}(Q), and hence the composite Ind ∘ Res does not change. Let

n_{H,Q} = |{K ≤ H, K^{g} = Q}_{H}|

denote the number of terms in the sum (2.10). Alternately, n_{H,Q} is the number of N_{G}(Q)- orbits in the set (G/H)^{Q}.

Similarly, the definitions of F_{∗} and F^{∗} imply that p_{K}F_{∗}(F^{∗}(a_{Q})) = 0, unless K ≤ H is G-conjugate to a subgroup of Q. Therefore

supp (E(RΓ_{G})(F_{∗}(F^{∗}(a_{Q}))) ⊆ supp(E_{Q}(a_{Q})) = supp(a).

By the Dress hypoelementary induction theorem [12, Theorem 7], [3, Cor. 3.5.8], there exist rational numbers {t_{H} | H ∈ 𝒢_{p}^{1}}, such that every element u ∈ A(R[N_{G}(Q)/Q]) satisfies the equation

(2.11)  u = ∑_{H∈𝒢_{p}^{1}} t_{H} Ind_{W_{H}(Q)}^{W_{G}(Q)} (Res_{W_{H}(Q)}^{W_{G}(Q)}(u))

where t_{H} = 0 unless Q ≤ H, and W_{G}(Q) = N_{G}(Q)/Q as usual. We will only need the induction result for elements in the subgroup K_{0}(R[N_{G}(Q)/Q]) ⊂ A(R[N_{G}(Q)/Q]). The proof shows that the general formula follows from the one for u = [R], the unit in the Dress ring (compare [17, Theorem 3.10]). We observe that the Dress formula only uses the p-hypoelementary subgroups of N_{G}(Q)/Q, and since Q ∈ ℱ any such subgroup has the form H/Q, where Q ◁ H and H ∈ 𝒢_{p}^{1}.

Moreover, we can assume that the coefficients {t_{H}} in (2.11) are invariant under conjugation, meaning that t_{H^{g}} = t_{H} for all g ∈ G. This follows by starting with the Dress induction formula for the unit [R] ∈ A(R[G]), where the inductions and restrictions from conjugate subgroups are equal, and then obtaining the formula for A(R[N_{G}(Q)/Q]) by restriction to R[N_{G}(Q)], followed by a generalized restriction to R[N_{G}(Q)/Q] in the sense of [16, 1.A.8].

We now define

b = a − ∑_{H∈𝒢_{p}^{1}} \frac{t_{H}}{n_{H,Q}} Ind_{H}^{G}(Res_{H}^{G}(a))

for a = E_{Q}(a_{Q}) ∈ K_{0}(RΓ_{G}), and note that

S(RΓ_{G})( ∑_{H∈𝒢_{p}^{1}} \frac{t_{H}}{n_{H,Q}} Ind_{H}^{G}(Res_{H}^{G}(a))) = ∑_{H∈𝒢_{p}^{1}} \frac{t_{H}}{n_{H,Q}} F_{∗}(F^{∗}(S(RΓ_{G})(a)))

by formula (2.9). However, by formulas (2.10) and (2.11) applied to u = a_{Q}, we have

S_{Q}(b) = S_{Q}(a) − ∑_{H∈𝒢_{p}^{1}} \frac{t_{H}}{n_{H,Q}} p_{Q}F_{∗}(F^{∗}(S(RΓ_{G})(a))) = 0,

and hence supp(b) < supp(a). By our inductive assumption, there exist rational numbers {z_{K}} such that

b = ∑_{K∈𝒢_{p}^{1}} z_{K} Ind_{K}^{G}(Res_{K}^{G}(b)).

By substituting the formula defining b into this expression, we obtain terms of the form

(Ind_{K}^{G} ∘ Res_{K}^{G} ∘ Ind_{H}^{G} ∘ Res_{H}^{G})(a)

for p-hypoelementary subgroups H and K. However, we can use the Mackey double coset formula to express Res_{K}^{G} ∘ Ind_{H}^{G} as a sum of terms of the form

Ind_{^{g}H∩K}^{K} ∘ c_{g} ∘ Res_{H∩K^{g}}^{H} .

Since these terms will be applied to Res_{H}(a), and conjugation acts as the identity on K_{0}(RΓ_{G}), the internal conjugations can be omitted. We have now obtained the desired result

a = ∑_{H∈𝒢_{p}^{1}} r_{H} Ind_{H}^{G}(Res_{H}^{G}(a)),

for any a ∈ K_{0}(RΓ_{G}) ≅ K_{0}(RE_{G}). Note that when this formula is applied to an element in the Burnside quotient Green ring A_{K}, it says that A_{K} is rationally generated by induction from the p-hypoelementary subgroups of G. □

3. Oliver’s actions on finite acyclic complexes

In this section, let R = ℤ̂_{p} or R = ℤ/p, for some prime p. We prove a result about the finiteness obstruction σ̃(C) ∈ K̃_{0}(RΓ) of a chain complex C over the orbit category (with respect to the family of p-subgroups of G), which is weakly homology equivalent to a finite projective chain complex. This follows from a more direct result about modules over the orbit category which have finite projective resolutions. As an application of these observations, we give an alternative approach to R. Oliver’s constructions of finite mod-p acyclic complexes.

Given a finite G-CW-complex X, there is an associated finite cellular chain complex

C(X^{?}; R): 0 → R[X_{n}^{?} ] → ⋯ → R[X_{1}^{?} ] → R[X_{0}^{?} ] → 0

of RΓ_{G}-modules, where X_{i} denotes the set of i-cells in X. An RΓ -module of the form R[G/H^{?} ] is not projective in general, but it has always a finite projective resolution (see Example 2.1). Recall that a weak homology equivalence between chain complexes over RΓ is a chain map inducing an isomorphism on homology (see [20, §11]).

Lemma 3.1. The complex C(X^{?}; R) is weakly homology equivalent to a finite projective complex P.

Proof. For each k ≥ 0, we have the k-skeleton X^{(k)} of X, which is a G-CW subcomplex, and a short exact sequence

0 → C(X^{(k−1)}^{?}; R) → C(X^{(k)}^{?}; R) → D^{(k)} → 0

of RΓ -module chain complexes, for k ≥ 1. The relative cellular complex

D^{(k)} = C((X^{(k)}, X^{(k−1)})^{?}; R) = R[X_{k} ^{?} ],

where we regard the module R[X_{k} ^{?} ] as a chain complex concentrated in degree k. For each k ≥ 0, we pick a finite projective resolution f^{(k)} : P^{(k)} → R[X_{k} ^{?} ], and regard P^{(k)} as a chain complex starting in degree k. The map f^{(k)} then gives a weak homology equivalence P^{(k)} → D^{(k)}. By induction on k and standard homological algebra (see [20, 11.2(c)]), we obtain a weak homology equivalence f: P → C(X^{?}; R) with P = ⨁ P^{(k)}. □

The obstruction for replacing a weak homology equivalence f: P → C(X^{?}; R) with a finite free chain complex (in the same chain homotopy type) is an element

σ̃(X) ∈ K̃_{0}(RΓ)

in the projective class group, defined as the image of the Euler characteristic

σ(X) = ∑(−1)^{i}[P_{i}] ∈ K_{0}(RΓ).

Note that this obstruction is defined for any finite G-CW complex X, so it is defined for finite G-sets as well (considered as G-CW complexes of dimension zero).

By uniqueness of projective resolutions (up to chain homotopy equivalence), the Euler characteristic σ(X), and hence the finiteness obstruction σ̃(X), is independent of the choice of projective complex P weakly homology equivalent to C(X^{?}; R). In particular, the proof of Lemma 3.1 shows that

(3.2)  σ(X) = ∑(−1)^{k}σ(X_{k}) ∈ K_{0}(RΓ),

where σ(X_{k}) is (by definition) the Euler characteristic of any finite projective resolution P^{(k)} for the module R[X_{k} ^{?} ]. The obstruction σ̃(X) = 0 if and only if there is a finite free chain complex with a weak homology equivalence to C(X^{?}; R).

We now recall a description of the Burnside ring B(G), due to tom Dieck [9, p. 239]. In this description, B(G) is the set of equivalence classes of finite G-CW complexes, with X ∼ Y if and only if χ(X^{H}) = χ(Y^{H}) for all subgroups H ≤ G. The addition is disjoint union and the multiplication is cartesian product. The additive identity is the empty set, and the additive inverse −[X] is represented by Z × X, for any finite complex Z with χ(Z) = −1 and trivial G-action.

If X is a finite G-CW complex, and {X_{k}} denotes the finite G-sets of k-cells, then the relation

[X] = ∑(−1)^{k}[X_{k}] ∈ B(G)

follows immediately from the definition above. Now this relation and formula (3.2) shows that σ(X) = σ(Y) ∈ K_{0}(RΓ) whenever χ(X^{H}) = χ(Y^{H}), for all subgroups H ≤ G.

The main result of this section is the following improvement:

Theorem 3.3. Let X and Y be two G-CW-complexes such that χ(X^{H}) = χ(Y^{H}) for every p-hypoelementary subgroup H in G, then σ(X) = σ(Y) ∈ K_{0}(RΓ_{G}).

As an application, we have a useful embedding result:

Corollary 3.4. Let X be a finite G-CW complex with the property that χ(X^{H}) = 1 for every H ∈ G_{p}^{1}. Then there exists a finite G-CW complex Y including X as a subcomplex such that

(i) Y \X only has cells with prime power stabilizers.

(ii) Y^{K} is mod p acyclic for every p-subgroup K.

Proof. Let R = ℤ/p. By Theorem 3.3, σ(X) = σ(pt). By attaching orbits of cells with stabilizers Q ∈ ℱ, we can also assume that the chain complex C := C(X^{?}; R) of the G-CW-complex X is n-dimensional, (n−1)-connected for n large, and has a single nontrivial homology H_{n}(C) = M in positive dimensions. This process does not change the finiteness obstruction, so we have σ̃(X) = σ̃(pt).

Since H_{0}(C(X^{?}; R)) = \underline{R} has a finite projective resolution, the exact sequence

0 → M → C_{n} → C_{n−1} → ⋯ → C_{0} → \underline{R} → 0

implies that Ext_{RΓ}^{k} (M, N) = 0, for all RΓ -modules N, if k is suffficiently large. Hence the RΓ -module M also has finite projective resolution and σ(X) = (−1)^{n}[M] + [\underline{R}], by [20, 11.9]. But σ(pt) = [\underline{R}], so the relation σ̃(X) = σ̃(pt) implies that σ̃(M) = 0, implying that M has a finite free resolution over RΓ. This shows that we can add more cells with stabilizers Q ∈ ℱ to kill the remaining homology on X and obtain a mod p acyclic complex satisfying the above properties. □

Before giving the proof of Theorem 3.3, we need some preparation. Recall that there is map called the linearization map from B(G) to the Green ring A(RG). The linearization map

Lin: B(G) → A(RG)

is defined as the linear extension of the assignment [X] → [RX] where RX denotes the permutation module with basis given by a finite G-set X. The linearization map is determined as follows:

Lemma 3.5 (Conlon). For a G-CW complex X, the class Lin_{R}([X]) = 0 if and only if χ(X^{H}) = 0 for every subgroup H ∈ 𝒢_{p}^{1}.

Proof. This is due to Conlon (see [6], or [3, Theorem 3.5.5]). The “if” direction is a special case of [12, Theorem 7]. The “only if” direction is the statement that the linearization map B(H) → A(RH) is injective for all H ∈ 𝒢_{p}^{1} (this also holds for R = ℤ by [12, Prop. 9.6]). □

Note that to prove Theorem 3.3, it is enough to prove it for G-sets X and Y satisfying the property that |X^{H}| = |Y^{H}| for all H ∈ 𝒢_{p}^{1}. By Conlon’s theorem, two such G-sets will then have isomorphic permutation modules RX ≅ RY.

Remark 3.6. If Q ◁ H, for some p-subgroup Q, then H/Q ∈ 𝒢_{p}^{1} if and only if H ∈ 𝒢_{p}^{1}. We may apply this remark to the N_{G}(Q)/Q-sets X^{Q} and Y^{Q}. By Conlon’s Theorem, the permutation modules R[X^{Q}] and R[Y^{Q}] will be isomorphic as R[N_{G}(Q)/Q]-modules, for every p-subgroup Q, since |(X^{Q})^{H/Q}| = |X^{H}| for all H/Q ≤ N_{G}(Q)/Q with H/Q ∈ 𝒢_{P}^{1}.

The proof of Theorem 3.3. We are considering modules over the orbit category Γ relative to the family ℱ of all p-subgroups in G. If X and Y are finite G-sets such that RX ≅ RY as RG-modules, then we wish to show that σ(X) = σ(Y). The argument will proceed in the following two steps:

(i) If G is p-hypoelementary, and RX ≅ RY as RG-modules, then we will show that R[X^{?} ] ≅ R[Y ^{?} ] as RΓ -modules.

(ii) We reduce to p-hypoelementary groups by applying Corollary 2.5.

To establish step (i) we now assume that G ∈ 𝒢_{p}^{1}. Since any subgroup of a phypoelementary group is also p-hypoelementary, we see that |X^{H}| = |Y^{H}| for all H ≤ G by Lemma 3.5. This shows that X ≅ Y as G-sets, and finishes step (i).

For any finite group G, we conclude by step (i) that Res_{H}^{G}(R[X^{?} ]) ≅ Res_{H}^{G}(R[Y ^{?} ]), for all H ∈ 𝒢_{p}^{1}, and therefore Res_{H}^{G}(σ(X)) = Res_{H}^{G}(σ(Y)), for all H ∈ 𝒢_{p}^{1}. By Corollary 2.5, we have σ(X) = σ(Y) ∈ K_{0}(RΓ). □

We remark that step (i) above only holds if G is p-hypoelementary. In general, given two G-sets X and Y such that RX ≅ RY as RG-modules, we can not conclude that R[X^{?} ] ≅ R[Y ^{?} ] as RΓ -modules, even though R[X^{Q}] ≅ R[Y^{Q}] as R[N_{G}(Q)/Q]-modules for every Q ∈ ℱ. In other words, the Dress detection result (Corollary 2.5) does not extend to A(RΓ_{G}). Here is an explicit example.

Example 3.7. Let G = S_{3}, R = ℤ/2 and ℱ be the family of all 2-subgroups in G. Let X = [G/1] + 2[G/G] and Y = 2[G/C_{2}] + [G/C_{3}]. Except for G, all subgroups of G are 2-hypoelementary. It is easy to see that |X^{K}| = |Y^{K}| for all K ≤ G and K ≠ G. So, RX ≅ RY as RG-modules and σ(X) = σ(Y) by Theorem 3.3. Note that the modules R[G/1^{?} ], R[G/C_{2}^{?} ], and R[G/C_{3}^{?} ] are all projective as RΓ -modules, but R[G/G^{?} ] is not since G does not have a normal Sylow 2-subgroup (see [25, Lemma 2.5]). Therefore, we can not have an isomorphism R[X^{?} ] ≅ R[Y ^{?} ], otherwise R[G/G^{?} ] would be projective.

As an application of Theorem 3.3, we will prove the following theorem of Oliver which is the key result in [21].

Theorem 3.8 (Oliver [21, Theorem 1]). Let G be a finite group not of p-power order, and φ a mod p resolving function for G. Then for any finite complex F with χ(F) = 1+φ(G), F is the fixed-point set of an action of G on some finite ℤ/p-acyclic complex.

A mod p resolving function is defined by Oliver in the following way:

Definition 3.9. A mod p resolving function for G is a super class function φ satisfying the following properties:

(i) |N_{G}(K)/K| divides φ(K) for all K ≤ G.

(ii) For any K ≤ G such that K ∈ G_{p}^{1}, we have ∑_{K≤L} φ(L) = 0.

We will give alternate description of mod p resolving functions. Note that there is a commutative diagram

[figure: 0; B(G); ρ; C(G); ψ; θ; Obs(G); 0; 0; B(G); η; C(G); γ; θ^{−1}; Obs(G); 0]

where B(G) denotes the Burnside ring of finite G-sets, C(G) denote the group of super class function, and

Obs(G) = ⨁_{K≤_{G}G} ℤ/|W_{G}(K)|ℤ.

The maps in the diagram are defined as follows: the map ρ is the mark homomorphism [11] defined by ρ(G/K)(L) = |(G/K)^{L}|, and η is defined by η([G/K])(L) = |W_{G}(K)| if K and L are conjugate to each other, and zero otherwise. The homomorphism γ is defined as the direct sum of the mod |W_{G}(K)| reductions, and ψ = γ ∘ θ. The map θ is an invertible transformation such that

θ(f)(K) = ∑_{K≤L} μ(K, L)f(L) and θ^{−1}(f)(K) = ∑_{K≤L} f(L).

Here μ(K, L) denotes the Möbius function for the poset of subgroups of G. More details about this diagram can be found in [7]. We have the following observation:

Lemma 3.10. A super class function φ is a mod p resolving function if and only if θ^{−1}(φ) is in the image of ρ and θ^{−1}(φ)(K) = 0 for all K ∈ 𝒢_{p}^{1}.

Proof. This follows form the above commuting diagram and from the definition of mod p resolving functions. □

Remark 3.11. Given F, and any group G not of p-power order, Oliver concludes in [21, Corollary, p. 162] that there exists a finite ℤ/p-acyclic G-CW complex with X^{G} = F if and only if

χ(F) ≡ 1 mod m_{p}(G),

where m_{p}(G) is the greatest common divisor of the integers {φ(G)} over all mod p resolving functions for G. The existence of mod p resolving functions for G is completely analyzed by Oliver in [21, Theorem 4], which gives the explicit characterization: (i) m_{p}(G) = 0 if and only if G ∈ 𝒢_{p}^{1}, (ii) m_{p} = 1 if G ∉ 𝒢_{p}, and (iii) m_{p} is the product of the distinct primes q such that G ∈ 𝒢_{p}^{q}, q > 1.

The proof of Theorem 3.8. Let φ be a mod p resolving function φ, and F a finite complex such that χ(F) = 1 + φ(G). Then by Lemma 3.10, the super class function f = θ^{−1}(φ) in the image of ρ, and φ(G) = f(G) so that χ(F) = 1 + f(G). Notice that 1 + f is also in the image of ρ: B(G) → C(G), since ρ([G/G])(K) = 1 for all K ≤ G.

From tom Dieck’s description [9, p. 239] of B(G), there exists a finite G-CW complex X with the properties:

(i) χ(X^{K}) = 1 + f(K) for every K ≤ G, (ii) χ(X^{H}) = 1 for all H ∈ 𝒢_{p}^{1}, and (iii) X^{G} = F

Property (ii) follows from the definition of θ, since φ = θ(f) is a mod p resolving function. To obtain property (iii), start with any finite G-CW complex X_{1} satisfying property (i) and let X_{0} = X_{1} \ U, where U is an open G-invariant regular neighbourhood of X_{1}^{G}, obtained by an equivariant triangulation of X_{1}. Let X = X_{0} ⊔ F. Note that X_{0} and therefore X has the structure of a finite G-CW complex. Then χ(X_{0}^{K}) = χ(X_{1}^{K}) − χ(X_{1}^{G}), for all K ≤ G. It follows that χ(X^{K}) = χ(X_{1}^{K}), for all K ≤ G, and hence [X] = [X_{1}] ∈ B(G).

By Corollary 3.4, there exists a mod p-acyclic G-CW complex Y, containing X as a subcomplex, such that Y \X only has cells with prime power stabilizers. Since G is not of p-power order, it follows that Y^{G} = X^{G} = F, and this completes the proof. □

Remark 3.12. Oliver [21, §3] also determined which finite complexes F appear as the fixed-point set X^{G}, for finite contractible G-CW complexes X. An integral resolving function for G is a super class function in C(G) which is a mod p resolving function for all primes p. The set of integral resolving functions forms a group, and m(G) is defined as the greatest common divisor of the integers φ(G) over all integral resolving functions.

Assume that G is not of prime power order. Given an integral resolving function φ for G, and non-empty finite complex F such that χ(F) = 1+φ(G), there exists a finite G-CW complex X such that χ(X^{H}) = 1, for all H ∈ 𝒢_{p}^{1} and all primes p, and with X^{G} = F (as in the proof of Theorem 3.8 above).

Let Γ denote the orbit category of G with respect to the family ℱ of all p subgroups, for all primes p. By attaching orbits of cells with stabilizers Q ∈ ℱ, we can also assume that the chain complex C := C(X^{?}; R) of the G-CW-complex X is n-dimensional, (n−1)- connected for n large, with H_{0}(C) = \underline{ℤ} and has a single nontrivial homology H_{n}(C) = M in positive dimensions.

For each prime p, the homology modules H_{i}(C) ⊗ ℤ̂_{p} admit finite projective resolutions over ℤ̂_{p}Γ_{G}, so by [15, Prop. 3.11] the homology modules H_{i}(C) admit finite projective resolutions over ℤΓ_{G}. Therefore, the finiteness obstruction σ(X) is defined, and

σ̃(X) = (−1)^{n}[M] + [\underline{ℤ}] ∈ K̃_{0}(ℤΓ_{G})

by [20, 11.9]. We call X a G-resolution of F, and define γ_{G}(F, X) := σ̃(X), following Oliver [21, §3]. Then define

γ_{G}(F) ∈ K̃_{0}(ℤΓ_{G})/ℬ(G)

to be the image of γ_{G}(X, F)), for any G-resolution X of F, where

ℬ(G) = {γ_{G}(pt, X) | X is a G-resolution of F = pt}.

Then γ_{G}(F) is well-defined, as in [21, Prop. 5]. If χ(F) = 1, and X is a G-resolution for F, then X/F is a G-resolution for (X/F)^{G} = pt, and hence γ_{G}(F) = 0 whenever χ(F) = 1. It follows as in [21, Theorem 3] that χ(F_{1}) = χ(F_{2}) implies γ_{G}(F_{1}) = γ_{G}(F_{2}). Since γ_{G}(F_{1} ∨ F_{2}) = γ_{G}(F_{1}) + γ_{G}(F_{2}), we also have the conclusion of [21, Corollary 5]. Let n_{G} denote the greatest common divisor of the integers {χ(F) − 1} as F varies over all finite complexes with χ(F) ≡ 1 mod m(G) and γ_{G}(F) = 0. Then F is the fixed point set of a finite contractible G-CW complex if and only if χ(F) ≡ 1 mod n_{G}.

It might be interesting to continue the study of ℬ(G) over the orbit category, in analogy with Oliver [22].

4. Acyclic permutation complexes

Let G be a discrete group. We say that X is a G-complex if X is a CW-complex with a G-action on it in a such a way that G permutes the cells in X and if G fixes a cell, then it fixes it pointwise. Note that a G-CW-complex is a G-complex and conversely, every G-complex has a G-CW-complex structure. For G-complexes, we have the following theorem of tom Dieck (Chapter II, Proposition 2.7 in [10]):

Theorem 4.1. If G is a discrete group and f: X → Y is a G-map between G-CW-complexes which induces homotopy equivalences X^{H} → Y^{H} between the H-fixed subspaces for all subgroups H ≤ G, then f is itself a G-homotopy equivalence.

Recently, Kropholler and Wall [19] gave an algebraic version of this theorem. To introduce their theorem, we need to give more definitions.

Let R be a commutative ring and X be a G-set. As usual, we denote by RX, the based RG-permutation module with basis X where G acts by permuting the basis. An RG-module homomorphism f: RX → RY between two based permutation modules is called admissible if it carries the submodule R[X^{H}] into R[Y^{H}] for all subgroups H ≤ G. A chain complex of based RG-permutation modules

C: ⋯ → R[X_{n}] → R[X_{n−1}] → ⋯ → R[X_{1}] → R[X_{0}] → 0

is called a special G-complex if all the boundary maps are admissible. A chain map f: C → D between special G-complexes is a called an admissible G-map if for each i, the map f_{i}: C_{i} → D_{i} is an admissible map.

Theorem 4.2 (Kropholler-Wall [19]). Let f: C → D be an admissible G-map between special G-complexes. If f induces a chain homotopy equivalence between the H-fixed subcomplexes for all subgroups H ≤ G, then f is itself a chain homotopy equivalence.

Here by a H-fixed point complex, we mean the subcomplex

→ R[X_{n}^{H}] → R[X_{n−1}^{H}] → ⋯ → R[X_{1}^{H}] → R[X_{0}^{H}] → 0.

It is clear that when f: C → D is an admissible G-map, then for each H ≤ G, it induces a chain map between fixed point complexes.

Observe that a based permutation RG-module R[X] can be considered as a module over the orbit category in a natural way: let Γ = Or G denote the orbit category over all subgroups in G. Associated to a permutation RG-module R[X] with an R-basis X, there is an RΓ -module R[X^{?} ] which is a free RΓ -module. Note that if f: RX → RY is admissible, then it induces an RΓ -module map f: R[X^{?} ] → R[Y ^{?} ]. Conversely, given a map between free RΓ -modules f: R[X^{?} ] → R[Y ^{?} ], evaluation of f at 1 gives an admissible map f(1): RX → RY. This gives a natural equivalence between the following two categories:

(i) The category of based RG-permutation modules and admissible maps.

(ii) The category of free RΓ -modules and RΓ -module maps.

The equivalence of these categories gives an alternative proof for Theorem 4.2 using the orbit category.

Proof. Let f: C → D be a admissible G-map between special G-complexes. Under the natural equivalence explained above, we can consider f as a chain map between free chain complexes of RΓ -modules. The condition that f induces homotopy equivalences between the H-fixed subcomplexes for all H ≤ G gives that f(H): C(H) → D(H) is an homotopy equivalence for all H ≤ G. This gives, in particular, that the induced map on homology f_{∗}(H): H_{∗}(C(H)) → H_{∗}(D(H)) is an isomorphism for all H ≤ G. But, H_{∗}(C(H)) = H_{∗}(C)(H), so we get that f_{∗}: H_{∗}(C) → H_{∗}(D) is an isomorphism of RΓ -modules. Now, by a standard theorem in homological algebra, this implies that f: C → D is a chain homotopy equivalence as a chain map of RΓ -modules. Evaluating f at 1, we get the desired result. □

Our interpretation of next result will use the following version of Smith theory:

Theorem 4.3 (Symonds [24, Corollary 4.5]). Let P be a p-group, Γ = Or P, and R = ℤ̂_{p} denote the p-adic integers. If C is a chain complex of projectives over RΓ that is bounded above, such that ℤ/p ⊗_{ℤ} C(1) is exact, then C is split exact.

Proof. This is a slight generalization of Corollary 4.5 in Symonds [24] and the proof follows easily from the argument given in [24] (see also Section 6 of Bouc [1] for similar results). □

In [19], Kropholler and Wall also gave an alternative proof for a theorem of Bouc [2] about acyclic simplicial complexes (and extended the statement to special G-complexes). We will give a proof using the orbit category and Theorem 4.3. Recall that, a complex C of RG modules is called acyclic if has zero homology everywhere except at dimension zero and H_{0}(C) = R. Also note that a complex of RG-modules is called G-split if it admits a chain contraction.

Theorem 4.4 (Kropholler-Wall [19]). Let G be a finite group and let C be a finite dimensional special ℤG-complex. If C is ℤ-acyclic, then the augmented chain complex C̃ is G-split.

Proof. The augmented chain complex

C̃: 0 → C_{n} → ⋯ → C_{1} → C_{0} → ℤ → 0

is an exact sequence of ℤG-permutation modules. To show that C is G-split, we need to show that the short exact sequences

0 → Z_{i} → C_{i} → Z_{i−1} → 0

in C̃ are all split exact sequences of ℤG-modules. Since all the modules involved are free over ℤ, the extension classes Ext_{ℤG}^{1}(Z_{i−1}, Z_{i}) are detected by restriction to the Ext-groups Ext_{ℤ_{p}P}^{1}(Z_{i−1}, Z_{i}) where P is a Sylow p-subgroup of G. So, it is enough to assume that G = P is a p-group and show that ℤ_{p} ⊗_{ℤ} C̃ is split.

As before, we can consider the complex C̃ as a complex of ℤΓ -modules where Γ = Or G. This is a chain complex of the form

D : 0 → ℤ[X_{n}^{?} ] → ⋯ → ℤ[X_{1}^{?} ] → ℤ[X_{0}^{?} ] → ℤ[G/G^{?} ] → 0,

where all the modules are free ℤΓ -modules. Evaluation of D at 1 gives the augmented complex C̃. Since C is acyclic, the complex ℤ/p⊗_{ℤ} C̃ = ℤ/p⊗_{ℤ}D(1) is exact by universal coefficient theorem. So, by Theorem 4.3, we obtain that ℤ̂_{p} ⊗_{ℤ} D is split exact, hence its evaluation at 1, which is the complex ℤ̂_{p} ⊗_{ℤ} C̃, is also split exact. □

Remark 4.5. Kropholler and Wall [19, §5] observed that Oliver’s results on fixed point sets of finite contractible G-CW complexes combined with Theorem 4.4 imply a Dress induction statement. Here is a variant of that observation: let F be a finite complex with χ(F) = 1 + m_{p}(G), for some prime p. Then there is a finite mod p acyclic G-CW complex X with X^{G} = F. By the mod p version of Theorem 4.4, the augmented chain complex D of C(X^{?}; R), for R = ℤ/p, is split over the orbit category and hence its evaluation C̃ = D(G/1) is G-split. This gives the relation that m_{p}(G) · [R] is a linear combination in A(RG) of permutation modules R[G/H], with H < G proper subgroups. This suggests that m_{p}(G) should be the optimal denominator in the Dress rational hypoelementary induction Theorem [12, Theorem 7].

It should also be pointed out that this implication is circular, since the proof of Oliver’s results involves directly or indirectly the same ingredients as Dress’s theorem.

Some other nice applications of Theorem 4.4 are given in [19]. One of them extends a result of Floyd [14, Theorem 2.12].

Theorem 4.6 (Theorem 6.1, [19]). Let G be a locally finite group and let X be a finite dimensional acyclic G-CW complex. Then, the complex X/G is acyclic.

Proof. We outline the steps of the argument given in [19]. Since a locally finite group is the directed union of its finite subgroups, it is enough to do the case where G is finite. Then C(X; ℤ) is ℤG-split, implying that the chain complex C(X; ℤ) ⊗_{ℤG}ℤ is also acyclic by Theorem 4.4. But this complex is isomorphic to the chain complex of X/G. □

References

[1] S. Bouc, Résolutions de foncteurs de Mackey, Group representations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 31–83.

[2] , Le complexe de chaînes d’un G-complexe simplicial acyclique, J. Algebra 220 (1999), 415– 436.

[3] , Burnside rings, Handbook of algebra, Vol. 2, North-Holland, Amsterdam, 2000, pp. 739–804.

[4] N. Brady, I. J. Leary, and B. E. A. Nucinkis, On algebraic and geometric dimensions for groups with torsion, J. London Math. Soc. (2) 64 (2001), 489–500.

[5] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin, 1967.

[6] S. B. Conlon, Structure in representation algebras, J. Algebra 5 (1967), 274–279.

[7] O. Coşkun and E. Yalçın, A Tate cohomology sequence for generalized Burnside rings, J. Pure Appl. Algebra 213 (2009), 1306–1315.

[8] C. W. Curtis and I. Reiner, Methods of representation theory. Vol. II, John Wiley & Sons Inc., New York, 1987, With applications to finite groups and orders, A Wiley-Interscience Publication.

[9] T. t. Dieck, The Burnside ring of a compact Lie group. I, Math. Ann. 215 (1975), 235–250.

[10] , Transformation groups, Arch. Math. (Basel) 8 (1987), x+312.

[11] A. W. M. Dress, A characterisation of solvable groups, Math. Z. 110 (1969), 213–217.

[12] , Contributions to the theory of induced representations, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 183–240. Lecture Notes in Math., Vol. 342.

[13] , Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math. (2) 102 (1975), 291–325.

[14] E. E. Floyd, Orbit spaces of finite transformation groups. I, Duke Math. J. 20 (1953), 563–567.

[15] I. Hambleton, S. Pamuk, and E. Yalçın, Equivariant CW-complexes and the orbit category, (arXiv:0807.3357v2 [math.AT]), 2008.

[16] I. Hambleton, L. R. Taylor, and E. B. Williams, Detection theorems for K-theory and L-theory, J. Pure Appl. Algebra 63 (1990), 247–299.

[17] , Dress Induction and the Burnside quotient Green ring, (arXiv:0803.3931[math.GR]), 2008.

[18] S. Jackowski, J. McClure, and B. Oliver, Homotopy classification of self-maps of BG via G-actions. II, Ann. of Math. (2) 135 (1992), 227–270.

[19] P. Kropholler and C. T. C. Wall, Group actions on algebraic cell complexes, (to appear: Publ. Mat. Univ. Autonoma Barcelona), 2008.

[20] W. Lück, Transformation groups and algebraic K-theory, Lecture Notes in Mathematics, vol. 1408, Springer-Verlag, Berlin, 1989, Mathematica Gottingensis.

[21] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177.

[22] , G-actions on disks and permutation representations, J. Algebra 50 (1978), 44–62.

[23] R. G. Swan, K-theory of finite groups and orders, Springer-Verlag, Berlin, 1970, Lecture Notes in Mathematics, Vol. 149.

[24] P. Symonds, Smith theory for algebraic varieties, Algebr. Geom. Topol. 4 (2004), 121–131 (electronic).

[25] , The Bredon cohomology of subgroup complexes, J. Pure Appl. Algebra 199 (2005), 261–298.

Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada

  E-mail address: hambleton@mcmaster.ca

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

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