EQUIVARIANT CW-COMPLEXES AND THE ORBIT CATEGORY

IAN HAMBLETON, SEMRA PAMUK, AND ERGÜN YALÇIN

Abstract. We give a general framework for studying G-CW complexes via the orbit category. As an application we show that the symmetric group G = S_{5} admits a finite G-CW complex X homotopy equivalent to a sphere, with cyclic isotropy subgroups.

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_{G} = Or_{F} G, where R is a commutative ring with unit. This theory was established by Bredon [6], tom Dieck [10] and Lück [20], and further developed by many authors (see, for example, Jackowski-McClure-Oliver [17, §5], Brady-Leary-Nucinkis [5], Symonds [34], [35], Grodal [14], Grodal-Smith [15]). In particular, the category of RΓ_{G}-modules is an abelian category with Hom and tensor product, and has enough projectives for standard homological algebra.

In this paper, we will study finite group actions on spheres with non-trivial isotropy, generalizing the approach of Swan [32] to the spherical space form problem through periodic projective resolutions. A finite group is said to have rank k if k is the largest integer such that G has an elementary abelian subgroup C_{p} ×⋯×C_{p} of rank k for some prime p. A rank 1 group G has periodic cohomology, and Swan showed that this was a necessary and sufficient condition for the existence of a finite free G-CW complex X, homotopy equivalent to a sphere.

The work of Adem-Smith [1] concerning free actions on products of spheres led to the following open problem:

Question. If G is a rank 2 finite group, does there exist a finite G-CW complex X ≃ S^{n} with rank 1 isotropy ?

If G is a finite p-group of rank 2, then there exist orthogonal linear representations V so that S(V) has rank 1 isotropy (see [12]). If G is not of prime power order, representation spheres with rank 1 isotropy do not exist in general: a necessary condition is that G has a p-effective character for each prime p dividing |G| (see [18, Thm. 47]). In [18, Prop. 48] it is claimed that this condition is also sufficient for an affirmative answer to the G-CW question above, but the discussion on [18, p. 831] does not provide a construction for X.

Date: Mar. 30, 2010 (revision 2).

Research partially supported by NSERC Discovery Grant A4000. The third author is partially supported by TÜBİTAK-BDP and TÜBA-GEBİP/2005-16.

Our main result concerns the first non-trivial case: the permutation group G = S_{5} of order 120, which has rank 2 but no linear action with rank 1 isotropy on any sphere, although it does admit p-effective characters for p = 2, 3, 5.

Theorem A. The permutation group G = S_{5} admits a finite G-CW complex X ≃ S^{n}, such that X^{H} ≠ ∅ implies that H is a rank 1 subgroup of 2-power order.

Remark 1.1. It is an interesting problem for future work to decide if the group G = S_{5} can act smoothly on S^{n} with rank 1 isotropy.

In order to prove this result we develop further techniques over the orbit category, which may have some independent interest. A well-known theorem of Rim [29] shows that a module M over the group ring ZG is projective if and only if its restriction Res_{P}^{G} M to any p-Sylow subgroup is projective. Over the orbit category we have a similar statement localized at p (see Theorem 3.9).

Theorem B. Let G be a finite group and let R = Z_{(p)}. Then an RΓ_{G}-module M has a finite projective resolution with respect to a family of p-subgroups if and only if its restriction Res_{P}^{G} M has a finite projective resolution over any p-Sylow subgroup P ≤ G.

Remark 1.2. For modules over the group ring RG, those having finite projective resolutions are already projective. Over the orbit category, these two properties are distinct.

Another useful feature of homological algebra over group rings is the detection of group cohomology by restriction to the p-Sylow subgroups. Here is an important concept in group cohomology (see for example [33]).

Definition 1.3. For a given prime p, we say that a subgroup H ⊆ G controls p-fusion provided that

(i) p ∤ |G/H|, and (ii) whenever Q ⊆ H is a p-subgroup, and there exists g ∈ G such that Q^{g} :=

g^{−1}Qg ⊆ H, then g = ch where c ∈ C_{G}(Q) and h ∈ H.

One reason for the importance of this definition is the fact that the restriction map

H^{∗}(G; 𝔽_{p}) → H^{∗}(H; 𝔽_{p})

is an isomorphism if and only if H controls p-fusion in G (see [25], [33]). We have the following generalization (see Theorem 5.1) for functors of cohomological type over the orbit category (with respect to any family F).

Theorem C. Let G be a finite group, R = Z_{(p)}, and H ≤ G a subgroup which controls p-fusion in G. If M is an RΓ_{G}-module and N is a cohomological Mackey functor, then the restriction map

Res_{H}^{G}: Ext_{RΓ_{G}}^{n}(M, N) → Ext_{RΓ_{H}}^{n}(Res_{H}^{G} M, Res_{H}^{G} N)

is an isomorphism for n ⩾ 0, provided that the centralizer C_{G}(Q) of any p-subgroup Q ≤ H, with Q ∈ F, acts trivially on M(Q) and N(Q).

The construction of the G-CW complex X for G = S_{5} and the proof of Theorem A is carried out in Section 9. We first construct finite projective chain complexes C^{(p)} over the orbit categories RΓ_{G}, with R = Z_{(p)}, separately for the prime p = 2, 3, 5 dividing |G|. In each case, the isotropy family F consists of the rank 1 subgroups of 2-power order in G.

The chain complexes C^{(p)} all have the same dimension function (see Definition 8.2). We prescribe a non-negative function \underline{n}: F → Z, with the property that \underline{n}(K) ≤ \underline{n}(H) whenever H is conjugate to a subgroup of K. Then, by construction, each complex C^{(p)} has the R-homology of an \underline{n}-sphere: for each K ∈ F, the complexes C^{(p)}(K) have homology H_{i} = R only in two dimensions i = 0 and i = \underline{n}(K). In other words, the complexes C^{(p)} are algebraic versions of tom Dieck’s homotopy representations [10, II.10].

In the case p = 2, we start with the group H = S_{4} acting by orthogonal rotations on the 2-sphere. A regular H-equivariant triangulation of an inscribed cube or octahedron gives a finite projective chain complex over RΓ_{H}. Then we use Proposition 6.4, a chain complex version of Theorem C, to lift it to a finite projective complex over RΓ_{G}. For p = 3 and p = 5, the p-rank of S_{5} is 1, and there exists a periodic complex over the group ring RG (see Swan [32, Theorem B]). We start with a periodic complex over RG and add chain complexes to this complex, for every nontrivial subgroup K ∈ F, to obtain the required complex C^{(p)} over RΓ_{G}.

We use the theory of algebraic Postnikov sections by Dold [11] to glue the complexes together to form a finite projective ZΓ_{G} chain complex (see Section 6). We complete the chain complex construction by varying the finiteness obstruction to obtain a complex of free ZΓ_{G}-modules, and then we prove a realization theorem (see Section 8) to construct the required G-CW complex X ≃ S^{n}.

Throughout the paper, a family of subgroups will always mean a collection of subgroups which is closed under conjugation and taking subgroups. Also, unless otherwise stated, all modules are finitely generated.

Acknowledgement. The authors would like to thank the referee for many valuable criticisms and suggestions. The third author would also like to thank McMaster University for the support provided by a H. L. Hooker Visiting Fellowship, and the Department of Mathematics & Statistics at McMaster for its hospitality while this paper was written.

2. Modules over small categories

Our main source for the material in this section is Lück [20, §9, §17] (see also [10, §I.10, §I.11]). We include it here for the convenience of the reader.

Let R be a commutative ring. We denote the category of R-modules by R-Mod. For a small category Γ (i.e., the objects Ob(Γ) of Γ form a set), the category of right RΓ - modules is defined as the category of contravariant functors Γ → R-Mod, where the objects are functors M(−): Γ → R-Mod and morphisms are natural transformations. Similarly, we define the category of left RΓ -modules as the category of covariant functors N(−): Γ → R-Mod. We denote the category of right RΓ -modules by Mod-RΓ and the category of left RΓ -modules by RΓ -Mod.

The category of covariant or contravariant functors from a small category to an abelian category has the structure of abelian category which is object-wise induced from the abelian category structure on abelian groups (see [23, Chapter 9, Prop. 3.1]). Hence the category of RΓ -modules is an abelian category where the notions submodule, quotient module, kernel, image, and cokernel are defined object-wise. The direct sum of RΓ - modules is given by taking the usual direct sum object-wise.

Example 2.1. The most important example for our applications is the orbit category of a finite group. Let G be a finite group and let F be a family of subgroups of G which is closed under conjugation and taking subgroups. The orbit category Or(G) is the category whose objects are subgroups H of G or coset spaces G/H of G, and the morphisms Mor(G/H, G/K) are given by the set of G-maps f: G/H → G/K.

The category Γ_{G} = Or_{F} G is defined as the full subcategory of Or(G) where the objects satisfy H ∈ F. The category of right RΓ_{G}-modules is the category of contravariant functors from Or_{F} G to R-modules. A right RΓ_{G}-module M is often called a coefficient system [35]. We will sometimes denote M(G/H) by M(H) if the group G is clear from the context. When F = {e}, RΓ_{G}-Mod is just the category of left RG-modules and Mod-RΓ_{G} is just the category of right RG-modules. □

Now, we will introduce the tensor product and Hom functors for modules over small categories. Let Γ be a small category and let M ∈ Mod-RΓ and N ∈ RΓ -Mod. The tensor product over RΓ is given by

M ⊗_{RΓ} N = ⨁_{x∈Ob(Γ)} M(x) ⊗ N(x)/ ∼

where ∼ is the equivalence relation defined by φ^{∗}(m)⊗n ∼ m⊗φ_{∗}(n) for every morphism φ : x → y. For RΓ -modules M and N, we mean by Hom_{RΓ} (M, N) the R-module of RΓ - homomorphisms from M to N. In other words,

Hom_{RΓ} (M, N) ⊆ ⨁_{x∈Ob(Γ)} Hom_{R}(M(x), N(x))

is the submodule satisfying the relations f(x) ∘ φ^{∗} = φ^{∗} ∘ f(y), for every morphism φ: x → y. We sometimes consider a second tensor product, namely the tensor product over R, which is defined for RΓ -modules M and N which are both left modules or both right modules. The tensor product M ⊗_{R} N is defined by the formula

[M ⊗_{R} N](x) = M(x) ⊗_{R} N(x)

on objects x ∈ Ob(Γ) and on morphisms, one has [M ⊗_{R} N](f) = M(f) ⊗_{R} N(f).

The tensor product over RΓ and Hom_{RΓ} are adjoint to each other. This can be described in the following way:

Proposition 2.2. Given two small categories Γ and Λ, the category of RΓ -RΛ-bimodules is defined as the category of functors Γ × Λ^{op} → R-Mod. For a right RΓ -module M, an RΓ -RΛ-bimodule B, and a right RΛ-module N, one has a natural transformation

Hom_{RΛ}(M ⊗_{RΓ} B, N) ≅ Hom_{RΓ} (M, Hom_{RΛ}(B, N)).

Proof. See [20, 9.21.] □

We will be using this isomorphism later when we are discussing induction and restriction.

2A. Free and finitely generated modules. For a small category Γ, a sequence

M′ → M → M′′

of RΓ -modules is exact if and only if

M′(x) → M(x) → M′′(x)

is exact for all x ∈ Ob(Γ). Recall that a module P in Mod-RΓ is projective if the functor

Hom_{RΓ} (P, −): Mod-RΓ → R-Mod

is exact. For an object x ∈ Γ, we define a right RΓ -module RΓ (?, x) by setting

RΓ (?, x)(y) = R Mor(y, x)

for all y ∈ Ob(Γ). Here, R Mor(y, x) denotes the free abelian group on the set of morphisms Mor(y, x) from y to x. As a consequence of the Yoneda lemma, we have

Hom_{RΓ} (RΓ (?, x), M) ≅ M(x).

So, for each x ∈ Ob(Γ), the module RΓ (?, x) is a projective module. When working with modules over small categories one uses the following notion of free modules.

Definition 2.3. Let Γ be a small category. A Ob(Γ)-set is defined as a set S together with a map β: S → Ob(Γ). We say a RΓ -module M is free if it is isomorphic to a module of the form

RΓ (S) = ⨁_{b∈S} RΓ (?, β(b))

for some Ob(Γ)-set S. A free module RΓ (S) is called finitely generated if the set S is finite.

Note that for every RΓ -module M, there is a free RΓ -module RΓ (S) and a map f: RΓ (S) → M such that f is surjective. We can take such a free module by choosing a set of generators S_{x} for the R-module M(x) for each x ∈ Ob(Γ), and then taking S as the Ob(Γ)-set which has the property β^{−1}(x) = S_{x}. A free module RΓ (S) which maps surjectively on M is called a free cover of M. A RΓ -module is called finitely generated if it has a finitely generated free cover.

It is clear from our description of free modules that an RΓ -module M is projective if and only if it is a direct summand of a free module. This shows that the module category of a small category has enough projectives. We will later give a more detailed description of projective RΓ -modules.

Example 2.4. For the orbit category Γ = Or(G), the free modules described above have a more specific meaning. For any subgroup K ≤ G, the RΓ -module RΓ (?, G/K) is given by

RΓ (?, G/K)(G/H) = R Mor(G/H, G/K) = R[(G/K)^{H}]

where R[(G/K)^{H}] is the free abelian group on the set of fixed points of the H action on G/K. Because of this, we denote the free module RΓ (?, G/K) by R[G/K^{?} ].

If F is a family of subgroups, and Γ_{G} = Or_{F} G, we obtain free RΓ_{G}-modules R[G/K^{?} ] by restriction whenever K ∈ F. The constant RΓ_{G}-module \underline{R} defined by \underline{R}(H) = R, for all H ∈ F, is just the restriction to RΓ_{G} of the module \underline{R} = R[G/G^{?} ]. This shows that the constant module \underline{R} is projective if G ∈ F. More generally, if K ∈ F, a non-empty fixed set

(G/K)^{H} = {gK | g^{−1}Hg ⊆ K} ≠ ∅

implies H ∈ F, since F is closed under conjugation and taking subgroups. Therefore, R[G/K^{?} ](H) = 0 for H ∉ F, whenever K ∈ F.

2B. Induction and Restriction. We now recall the definitions and terminology for these terms presented in Lück [20, 9.15]. Let Γ and Λ be two small categories. Given a covariant functor F: Λ → Γ, we define an RΛ-RΓ -bimodule

R(??, F (?)): Λ × Γ^{op} → R-Mod

on objects by (x, y) → R Hom(y, F (x)). We define the restriction map

Res_{F} : Mod-RΓ → Mod-RΛ

as the composition with F. The induction map

Ind_{F} : Mod-RΛ → Mod-RΓ

is defined by

Ind_{F} (M)(??) = M ⊗_{RΛ} R(??, F (?))

for every RΛ-module M. For every right RΓ -module N, the RΛ-module

Hom_{RΓ} (R(??, F (?)), N)

is the same as the composition Λ ⟶^{F} Γ ⟶^{N} R-Mod. So, by Proposition 2.2, we can conclude the following:

Proposition 2.5. Induction and restriction are adjoint functors: for any RΓ -module M and RΛ-module N, there is a natural isomorphism

Hom_{RΓ} (Ind_{F} M, N) = Hom_{RΛ}(M, Res_{F} N).

The induction functor respects direct sum, finitely generated, free, and projective but it is not exact in general. The restriction functor is exact but does not respect finitely generated, free, or projective in general.

Now we will define functors which are special cases of the restriction and induction functors. Let Γ be a small category. For x ∈ Ob(Γ), we define R[x] = R Aut(x) to be the group ring of the automorphism group Aut(x) and denote the category of right R[x]-modules by Mod-R[x]. Let Γ_{x} denote the full subcategory of Γ with single object x and let F: Γ_{x} → Γ be the inclusion natural transformation. The restriction functor associated to F gives a functor

Res_{x}: Mod-RΓ → Mod-R[x]

which is called the restriction functor. This functor behaves like an evaluation map Res_{x}(M) = M(x). In the other direction, the induction functor associated to F gives a functor

E_{x}: Mod-R[x] → Mod-RΓ

which is called the extension functor. For a R[x]-module M, we define E_{x}(M)(y) = M ⊗_{R[x]} R Mor(y, x) for every y ∈ Ob(Γ). They form an adjoint pair: for every R[x]- module M and an RΓ -module N, we have

Hom_{RΓ} (E_{x}M, N) ≅ Hom_{R[x]}(M, Res_{x} N).

By general properties of restriction and induction, the functor Res_{x} is exact and E_{x} takes projectives to projectives. In general, E_{x} is not exact and Res_{x} does not take projectives to projectives. But in some special cases, we can say more. For example, when Γ is free, i.e. R Mor(y, x) is a free R[x]-module for all y ∈ Γ, then it is easy to see that E_{x} is exact [20, 16.9].

Example 2.6. In the case of an orbit category Γ_{G} = Or_{F} G, we denote the extension function for H ∈ F simply by E_{H} and the restriction functor by Res_{H}. In this case, the automorphism group Aut(G/H) for H ∈ F is isomorphic to the quotient group N_{G}(H)/H. The isomorphism N_{G}(H)/H ≅ Aut(G/H) is given by the isomorphism nH → f_{n} where f_{n}(gH) = gn^{−1}H for n ∈ N_{G}(H) (see [10, Example 11.2]). This isomorphism takes right R[x]-modules to right R[N_{G}(H)/H]-modules, so given a right RΓ -module M, the evaluation at H ∈ F gives a right R[N_{G}(H)/H]-module.

It is easy to see that the morphism set Mor(G/K, G/H) is a free [N_{G}(H)/H]-set, so Or_{F} G is free in the above sense [20, Example 16.2]. Therefore, the functor E_{H} is exact and preserves projectives, whereas Res_{H} is exact but does not necessarily preserve projectives. For example, the module Z[G/G^{?} ] is free over Z Or(G) by definition, but Res_{H} Z[G/G^{?} ] = Z is not projective whenever N_{G}(H)/H ≠ 1.

2C. Inclusion and Splitting Functors. We will introduce two more functors. These are also special cases of induction and restriction, but they are defined through a bimodule rather than just a natural transformation F. We first describe these functors and then give their interpretations as restriction and induction functors.

Let Γ be an EI-category. By this, we mean that Γ is a small category where every endomorphism x → x is an isomorphism for all x ∈ Ob(Γ). This allows us to define a partial ordering on the set Iso(Γ) of isomorphism classes x̄ of objects x in Γ. For x, y ∈ Ob(Γ), we say x̄ ≤ ȳ if and only if Mor(x, y) ≠ ∅. The EI-property ensures that x̄ ≤ ȳ ≤ x̄ implies x̄ = ȳ.

For each object x ∈ Γ, and M ∈ Mod-R[x], the inclusion functor,

I_{x}: Mod-R[x] → Mod-RΓ

is defined by

I_{x}M(y) = \begin{cases}M ⊗_{R[x]} R Mor(y, x) & if ȳ = x̄ \\ \{0\} & if ȳ ≠ x̄. \end{cases}

In the other direction, we define the splitting functor

S_{x}: Mod-RΓ → Mod-R[x]

by S_{x}(M) = M(x)/M(x)_{s} where M(x)_{s} is the R-submodule of M(x) which is generated by the images of M(f): M(y) → M(x) for all f: x → y with x̄ ≤ ȳ and x̄ ≠ ȳ.

There is a RΓ -R[x]-bimodule B defined in such a way that the inclusion functor I_{x} can be described as M → Hom_{R[x]}(B, M) and the splitting functor S_{x} is the same as the functor M → M ⊗_{RΓ} B (see [20, page 171] for details). So (S_{x}, I_{x}) is an adjoint pair, meaning that

Hom_{R[x]}(S_{x}M, N) ≅ Hom_{RΓ} (M, I_{x}N)

for every RΓ -module M and R[x]-module N.

From general properties of induction and restriction, we can conclude that I_{x} is exact and S_{x} preserves projectives. Some of the other properties of these functors are listed in [20, Lemma 9.31]. It is interesting to note that the composition S_{x} ∘ E_{x} is naturally equivalent to the identity functor. Also, the composition S_{y} ∘ E_{x} is zero when x̄ ≠ ȳ. These are used to give a splitting for projective RΓ -modules.

Theorem 2.7. Let P be a finitely generated projective RΓ -module. Then

P ≅ ⨁_{x∈Iso(Γ)} E_{x}S_{x}(P).

Proof. For proof see [20, Corollary 9.40]. □

In the statement, the notation ⨁_{x∈Iso(Γ)} means that the sum is over a set of representatives x ∈ Ob(Γ) for x̄ ∈ Iso(Γ).

2D. Resolutions for RΓ -modules. Let Γ be an EI-category. For a non-negative integer l we define an l-chain c from x ∈ Ob(Γ) to y ∈ Ob(Γ) to be a sequence

c : x̄ = x̄_{0} < x̄_{1} < ⋯ < x̄_{l} = ȳ .

Define the length l(y) of y ∈ Ob(Γ) to be the largest integer l such that there exists an l-chain from some x ∈ Ob(x) to y. The length l(Γ) of Γ is max{l(x) | x ∈ Ob(Γ)}. Given an RΓ -module M, its length l(M) is defined by max{l(x) | M(x) ≠ 0} if M is not the zero module and l({0}) = −1.

We call a category Γ finite if Iso(Γ) and Mor(x, y) are finite for all x, y ∈ Ob(Γ). Denote by m(Γ) the least common multiple of all the integers | Aut(x)|.

Given an RΓ -module M, consider the map

ϕ: ⨁_{x∈Iso(Γ)} E_{x} Res_{x} M → M

where for each x ∈ Ob(Γ), the map ϕ_{x}: E_{x} Res_{x} M → M is the map adjoint to the identity homomorphism. It is easy to see that ϕ is surjective. Let

EM := ⨁_{x∈Iso(Γ)} E_{x} Res_{x} M

and let KM denote the kernel of ϕ: EM → M. Note that if x is an object with l(x) = l(M), then Res_{x} = S_{x} which also gives that

Res_{x} ϕ: Res_{x} E_{x} Res_{x} M → Res_{x} M

is an isomorphism. Note that this implies l(KM) < l(M) which allows one to proceed by induction and obtain the following:

Proposition 2.8. If Γ is finite EI-category, then every nonzero M has a finite resolution of the form

0 → EK^{t}M → ⋯ → EKM → EM → M → 0 .

where t = l(M).

Proof. See [20, 17.13 ]. Here K^{0}M = M and K^{s}M = K(K^{s−1}M). □

From the description above it is easy to see that

EK^{s}M := ⨁_{x∈Iso(Γ)} E_{x} Res_{x} K^{s}M

where Res_{x} K^{s}M is isomorphic to a direct sum of R[x]-modules

M(c) := M(x_{0}) ⊗_{R[x_{0}]} R Mor(x_{1}, x_{0}) ⊗_{R[x_{1}]} ⋯ ⊗_{R[x_{s−1}]} R Mor(x, x_{s−1})

over representatives in Ob(Γ) for all the chains of the form c : x̄ < x̄_{s−1} < ⋯ < x̄_{0} (see [20, 17.24]). Note that if Γ is a finite, free EI-category, then the resolution given in Proposition 2.8 will be a finite projective resolution if M(c) is projective as an R[x]-module for every chain c. This gives the following:

Proposition 2.9. Let M be RΓ_{G} module where Γ_{G} = Or_{F} G for some finite group G and R is a commutative ring such that |G| is invertible in R. Suppose also that M(H) is projective as an R-module for all H ∈ F. Then, M has a projective resolution with length less than or equal to l(Γ).

Proof. See [20, 17.31]. □

In particular, if R = Z_{(p)} with p ∤ |G| and if M is a RΓ -module such that M(H) is R-torsion free for all H ∈ F, then M has a finite projective resolution of length less than or equal to l(M).

3. The proof of Theorem B

The main result of this section is Theorem 3.9, which is an orbit category version of a well-known result of Rim [29]. We first collect some further information about induction and restriction for subgroups.

Let G be a finite group and let H be a subgroup of G. Given a family of subgroups F of G, we consider the orbit categories Γ_{G} = Or_{F} G and Γ_{H} = Or_{F} H, where the objects of Γ_{H} are orbits of H with isotropy in F_{H} = {K ≤ H | K ∈ F}. Let F: Γ_{H} → Γ_{G} be the functor which takes H/K to G/K and sends an H-map f: H/K → H/L to the induced G-map

Ind_{H}^{G}(f): G/K = G ×_{H} H/K → G ×_{H} H/L = G/L

for every K, L ∈ F_{H}. Note that if f is the map which takes eK to hL, then Ind_{H}^{G}(f)(gK) = ghL. The restriction and induction functors (see Proposition 2.5) associated to this functor gives us two adjoint functors

Res_{H}^{G}: Mod-Γ_{G} → Mod-Γ_{H}

and

Ind_{H}^{G}: Mod-Γ_{H} → Mod-Γ_{G}.

The restriction functor is defined as the composition with F. So, for a RΓ_{G}-module M, we have (Res_{H}^{G} M)(K) = M(K), for all K ∈ F_{H}. For the induced module we have the following formula:

Lemma 3.1. Let N be a RΓ_{H}-module and K ≤ G. Then,

(Ind_{H}^{G} N)(K) ≅ ⨁_{gH∈G/H, K^{g}≤H} N(K^{g})

where K^{g} = g^{−1}Kg.

Proof. For a (right) RΓ_{H}-module N, the induced module Ind_{H}^{G} N is defined as the direct sum

⨁_{L≤H} N(L) ⊗_{R} R Mor(G/K, G/L)

modulo the relations n ⊗ φf ∼ φ^{∗}(n) ⊗ f where n ∈ N(L), f ∈ Mor(G/K, G/L′) and φ = Ind_{H}^{G}(ϕ) for some ϕ : H/L′ → H/L. Every morphism G/K → G/L which satisfies the condition L ≤ H can be written as a composition φf_{g} where φ: G/K^{g} → G/L is induced from an H-map and f_{g}: G/K → G/K^{g} is given by xK → xgK^{g}, for some g ∈ G.

This shows that every element in the above sum is equivalent to an element of the form n ⊗ f_{g} where n ∈ N(K^{g}) and f_{g}: G/K → G/K^{g} is as above with K^{g} ≤ H. There is one summand for each gH satisfying K^{g} ≤ H. □

Note that we can also express the above formula by

(Ind_{H}^{G} N)(K) ≅ ⨁_{gH∈(G/H)^{K}} N(K^{g}).

If J ≤ K, then the argument above can be extended to show that restriction map

(Ind_{H}^{G} N)(K) → (Ind_{H}^{G} N)(J)

is given by the coordinate-wise restriction maps N(K^{g}) → N(J^{g}). Note that if gH ∈ (G/H)^{K}, then gH ∈ (G/H)^{J}. Similarly, the conjugation map

(Ind_{H}^{G} N)(K) → (Ind_{H}^{G} N)(^{x}K)

can be described by coordinate-wise conjugation maps. From these, it is easy to see that Ind_{H}^{G} \underline{R} ≅ R[G/H^{?} ]. A generalization of this argument gives the following:

Lemma 3.2. [35, Cor. 2.12]. Let G be a finite group and let H be a subgroup of G. For every RΓ_{G}-module M, we have Ind_{H}^{G} Res_{H}^{G} M ≅ M ⊗_{R} R[G/H^{?} ].

We also have the following formulas:

Lemma 3.3. Let G be a finite group and let H be a subgroup of G.

(i) For every K ≤ H, we have Ind_{H}^{G} R[H/K^{?} ] ≅ R[G/K^{?} ].

(ii) For every K ≤ G, we have Res_{H}^{G} R[G/K^{?} ] ≅ ⨁_{K\G/H} R[H/(H ∩^{g}K)^{?} ].

Proof. Part (i) follows from the fact that Ind_{H}^{G} Ind_{K}^{H} = Ind_{K}^{G} which is a consequence of a more general formula Ind_{F} Ind_{F′} = Ind_{F ∘F′}. We can prove this more general formula by using adjointness and the formula Res_{F′} Res_{F} = Res_{F ∘F′}. For (ii), observe that the definition of R[G/H^{?} ] can be extended to define a RΓ_{G}-module R[S^{?} ] for every G-set S, by taking

(3.4)  R[S^{?} ](G/K) = R Map_{G}(G/K, S)

for every K ∈ F, where Map_{G}(G/K, S) denotes the set of G-maps from G/K to S. For G-sets S and T, we have an isomorphism R[(S ⨆ T)^{?} ] ≅ R[S^{?} ]⊕R[T ^{?} ]. By the definition of restriction map, we get

(Res_{H}^{G} R[S^{?} ])(H/K) = R Map_{G}(G/K, S) = R Map_{H}(H/K, Res_{H}^{G} S).

It is easy to see that this induces an RΓ_{H}-module isomorphism

Res_{H}^{G} R[S^{?} ] ≅ R[(Res_{H}^{G} S)^{?} ].

Since

Res_{H}^{G}(G/K) ≅ ∐_{H\G/K} H/(H ∩^{g}K)

as G-sets, we obtain the formula given in (ii). □

Example 3.5. Let G = S_{5} be the symmetric group on {1, 2, 3, 4, 5} and H = S_{4} be the subgroup of symmetries that fix 5. Let C_{2} = ⟨(12)⟩ and C_{3} = ⟨(345)⟩. The formula in Lemma 3.3 (ii) gives

Res_{H}^{G} R[G/(C_{2} × C_{3})^{?} ] = R[H/C_{2} ^{?} ] ⊕ R[H/^{g}C_{3} ^{?} ]

where ^{g}C_{3} = ⟨(123)⟩. From this expression we obtain

R[G/(C_{2} × C_{3})^{?} ](G/C_{2}) ≅ R[H/C_{2} ^{?} ](H/C_{2}) ≅ R[N_{H}(C_{2})/C_{2}],

as an N_{H}(C_{2})/C_{2}-module, where N_{H}(C_{2}) = C_{2} × C_{2}. Note that N_{G}(C_{2}) = C_{2} × S_{3} and as an N_{G}(C_{2})/C_{2}-module R[G/(C_{2} × C_{3})^{?} ](G/C_{2}) is isomorphic to R[C_{2} × S_{3}/C_{2} × C_{3}]. □

We can give a more general formula for R[G/H^{?} ](G/K) as follows:

Lemma 3.6. Let G be a finite group, and H and K be two subgroups of G. Then, as an R[N_{G}(H)/H]-module

R[G/K^{?} ](G/H) ≅ ⨁_{v(H,K)} R[N_{G}(H)/N^{g}_{K}(H)]

where the sum is over the set v(H, K) of representatives of K-conjugacy classes of subgroups H^{g} such that H^{g} ≤ K.

Proof. This formula can easily be proved by first determining the orbits of N_{G}(H) action on (G/K)^{H} = {gK | H^{g} ≤ K}, and then by calculating the isotropy subgroups for each of these orbits. A similar computation can be found in the proof of Theorem 4.1 in [8]. □

Proposition 3.7. Both Res_{H}^{G} and Ind_{H}^{G} are exact and take projectives to projectives.

Proof. The fact that Res_{H}^{G} is exact and Ind_{H}^{G} preserves projectives follows from the general properties of restriction and induction functor associated to a natural transformation F. The fact that Ind_{H}^{G} is exact follows from the formula given in Lemma 3.1. Finally, to show that Res_{H}^{G} takes projective to projectives, it is enough to show it takes free modules to projective modules. An indecomposable free RΓ_{G}-module M is of the form R[G/K^{?} ] for some K ∈ F. By Lemma 3.3, Res_{H}^{G}(R[G/K^{?} ]) will be projective if H ∩^{g}K is in F for all HgK ∈ H\G/K. But this is always true since the family F is closed under conjugation and taking subgroups. □

A result of Rim [29] relates projectivity over the group ring ZG to projectivity over the p-Sylow subgroups.

Proposition 3.8 (Rim’s Theorem). Let G be a finite group, and M be a finitely generated ZG-module. Then M is projective over ZG if and only if Res_{P}^{G} M is projective over ZP for any p-Sylow subgroup P ≤ G.

Proof. A module M is ZG-projective if and only if Ext_{ZG}^{1}(M, N) = 0 for every ZG-module N. Therefore M is projective if and only if Z_{(p)} ⊗_{Z} M is projective over Z_{(p)}G for all primes p dividing the order of G.

For any p-Sylow subgroup P ≤ G, the permutation module R[G/P] ≅ R ⊕ N splits when R = Z_{(p)}. Therefore, if M is any RG-module, M ⊗_{R}R[G/P] ≅ M ⊕(M ⊗_{R}N). Since M ⊗_{R} R[G/P] ≅ Ind_{P}^{G} Res_{P}^{G} M, the projectivity of M is equivalent to the projectivity of Res_{P}^{G} M. □

Here is an orbit category version of this result.

Theorem 3.9 (Rim’s Theorem for the Orbit Category). Let G be a finite group and let M be a RΓ_{G}-module where R = Z_{(p)}. Suppose that F is a family of p-subgroups in G. Then M has a finite projective resolution if and only if Res_{P}^{G} M has a finite projective resolution for any p-Sylow subgroup P of G.

Proof. One direction is clear since Res_{P}^{G} is exact and takes projectives to projectives. For the other direction, we will give the proof by induction on the length l(M) of M. Without loss of generality, we can assume that M(H) is R-torsion free for all H ∈ F. Suppose M is a RΓ_{G}-module with l(M) = 0. Then, we can regard M as an RG-module. If Res_{P}^{G} M has a finite projective resolution, then Res_{P}^{G} M must be projective (see [20, page 348]). Then, by Rim’s theorem, M is a projective RG-module, hence has finite projective length.

Now, assume M is an RΓ_{G}-module with l(M) = s > 0. Let

0 → P_{n} → ⋯ → P_{0} → Res_{P}^{G} M → 0

be a projective resolution for Res_{P}^{G} M. We can assume that l(P_{i}) ⩽ s for all i. Then, for Q ∈ F with l(Q) = s, we have

S_{Q}P_{i} = Res_{Q} P_{i} = P_{i}(Q).

Since S_{Q} takes projectives to projectives, the resolution

0 → P_{n}(Q) → ⋯ → P_{0}(Q) → (Res_{P}^{G} M)(Q) → 0

is a finite projective resolution of (Res_{P}^{G} M)(Q) = M(Q) as an R[N_{P} (Q)/Q]-module. This gives that M(Q) is projective as an R[N_{P} (Q)/Q]-module.

Lemma 3.10. For every p-group Q, there is a p-Sylow subgroup P of G such that N_{P} (Q) is a p-Sylow subgroup of N_{G}(Q).

Proof. Let S be a p-Sylow subgroup of N_{G}(Q), and pick a p-Sylow subgroup P of G containing S. Since N_{P} (Q) = N_{G}(Q) ∩ P is a p-subgroup of N_{G}(Q), we have |N_{P} (Q)| ≤ |S|. But S ≤ P and S ≤ N_{G}(Q) implies S ≤ N_{P} (Q). Therefore S = N_{P} (Q). □

We can assume P is a p-Sylow subgroup which has this property. Then, by the p-local version of Rim’s theorem, we can conclude that M(Q) is projective as an R[N_{G}(Q)/Q]- module. Now, consider the map

ψ = (ψ_{Q}): ⨁_{Q∈Iso(Γ_{G}), l(Q)=s} E_{Q} ∘ Res_{Q} M → M

where ψ_{Q}: E_{Q} ∘ Res_{Q} M → M is the map adjoint to the identity map id: Res_{Q} M → Res_{Q} M. For every K ∈ F with l(K) = s, the induced map ψ(K) is an isomorphism. This is because

(E_{Q} ∘ Res_{Q} M)(K) = Res_{K} E_{Q} Res_{Q} M = S_{K}E_{Q} Res_{Q} M ≅ M(K)

if K is conjugate to Q and zero otherwise. So, we have l(coker ψ) < s. Therefore, there is a finitely generated projective RΓ_{G}-module P with l(P) < s, and a map α: P → M such that ψ ⊕ α is surjective. If K is the kernel of ψ ⊕ α, we get an exact sequence of RΓ_{G}-modules

0 → K → P ⊕ ⨁_{Q∈Iso(Γ_{G}), l(Q)=s} E_{Q} ∘ Res_{Q} M → M → 0

where the middle term is projective as an RΓ_{G}-module, and l(K) < s. Note that Res_{P}^{G} K must have a finite projective resolution by [20, Lemma 11.6]. So, by induction, K has a finite projective resolution, and hence M has a finite projective resolution as well. □

Remark 3.11. The inductive argument we use in the above proof is similar to the argument used by Lück to prove Proposition 17.31 in [20]. By this result, any module M over a finite EI-category Γ which has a finite projective resolution, admits a resolution of length ⩽ l(M) provided that M(x) is R-projective for all x ∈ Ob(Γ). □

It isn’t clear to us how to generalize Theorem 3.9 to integer coefficients. For R = Z_{(p)}, the following example shows that the result does not hold for an arbitrary family F.

Example 3.12. Let G = S_{5} and R = Z_{(2)} , and take F as the family of all 2-subgroups and 3-subgroups in G. Consider the RΓ_{G}-module M = R[G/(C_{2} × C_{3})^{?} ] where C_{2} and C_{3} are as in Example 3.5. It is clear that the restriction of M to a 2-Sylow subgroup is projective (since its restriction to H = S_{4} is already projective), but M does not have a finite projective resolution as an RΓ_{G}-module.

To see this, suppose that M has a finite projective resolution P ↠ M. Then, P(C_{3}) will be a finite projective resolution for M(C_{3}) over R[N_{G}(C_{3})/C_{3}]. This is because C_{3} = ⟨(123)⟩ is a maximal subgroup in F. This implies

M(C_{3}) ≅ R[S_{3} × C_{2}/C_{3} × C_{2}] ≅ R[C_{2}]

is projective as an R[N_{G}(C_{3})/C_{3}]-module. But,

R[N_{G}(C_{3})/C_{3}] = R[S_{3} × C_{2}/C_{3}] ≅ R[C_{2} × C_{2}],

and it is clear that R[C_{2}] is not projective as an R[C_{2} × C_{2}]-module. So, M does not have a finite projective resolution. □

On the other hand, the following holds for modules over orbit categories:

Proposition 3.13. Let G be a finite group, and F be a family of subgroups of G. Then, a ZΓ_{G}-module M has a finite projective resolution if and only if Z_{(p)} ⊗_{Z} M has a finite projective resolution over Z_{(p)}Γ_{G}, for all primes p dividing the order of G.

The proof of this statement follows from Propositions 4.4 and 4.5 in the next section. We end this section with some corollaries of Theorem 3.9.

Corollary 3.14. Let G be a finite group and R = Z_{(p)}. Suppose that F is a family of p-subgroups. Then, R[G/H^{?} ] has a finite projective resolution over RΓ_{G} if a p-Sylow subgroup of H is included in F.

Proof. If a p-Sylow subgroup of H is in F, then Res_{P}^{G} R[G/H^{?} ] is a free RΓ_{P} -module for any P ∈ Syl_{p}(G). So, by Theorem 3.9, it has a finite projective resolution. □

As a special case of this corollary, we obtain the following known result (see [4, 6.8], [35, 2.5 and p. 296], [17], [14]).

Corollary 3.15. Let G be a finite group and R = Z_{(p)}. Then, \underline{R} has a finite projective resolution over RΓ_{G} relative to the family of all p-subgroups of G.

Proof. This follows from \underline{R} = R[G/G^{?} ]. □

4. Mackey structures on Ext_{RΓ_{G}}^{∗}(M, N)

The notation and results of the previous sections will now be used to establish some structural and computational facts about the Ext-groups over the orbit category. Our main sources are Cartan-Eilenberg [7] and tom Dieck [10, §II.9] (see also [17], [14]).

We have seen that the category of right RΓ -modules has enough projectives to define the bifunctor

Ext_{RΓ}^{∗} (M, N) = H^{∗}(Hom_{RΓ} (P, N))

via any projective resolution P ↠ M (see [20, Chap. III, §17], [23, Chap. III.6]). The following property is also useful (see Lück [20, 17.21]).

Lemma 4.1. If Γ is a free EI-category, then Ext_{RΓ}^{∗} (E_{x}M, N) ≅ Ext_{R[x]}^{∗}(M, Res_{x} N).

Proof. Take a projective resolution P of M. Since Γ is free, the extension functor E_{x} is exact [20, 16.9]. In addition, E_{x} preserves projectives and is adjoint to the restriction functor Res_{x} by Proposition 2.5. Therefore

⋯ → E_{x}P_{n} → ⋯ → E_{x}P_{1} → E_{x}P_{0} → E_{x}M → 0

is a projective resolution of E_{x}M, and applying Hom over the orbit category gives

Ext_{RΓ}^{n} (E_{x}M, N) = H^{n}(Hom_{RΓ} (E_{x}P, N))
≅ H^{n}(Hom_{R[x]}(P, Res_{x} N)) = Ext_{R[x]}^{n}(M, Res_{x} N). □

In the rest of this section, we assume that Γ_{G} = Or_{F} G for a finite group G, where F is a family of subgroups in G. Note that Γ_{G} is both finite and free as an EI-category. If there are two groups H ≤ G, we use the notations Γ_{G} = Or_{F} G for the orbit category with respect to the family F, and Γ_{H} = Or_{F} H for the orbit category with respect to the family F_{H} = {H ∩ K | K ∈ F}.

Proposition 4.2. Let M and N be two ZΓ_{G}-modules, where M(H) is Z-torsion free for all H ∈ F. Then for every n > l(M), the groups Ext_{ZΓ_{G}}^{n}(M, N) are finite abelian, with exponent dividing some power of |G|.

Proof. This follows from the Lemma 4.1, Proposition 2.8, and the corresponding result for modules over finite groups. □

Note that the Ext-groups in lower dimensions are not finite in general. But, it is still true in all dimensions that the Ext-groups over ZΓ_{G} vanish if and only if they vanish over Z_{(p)}Γ_{G}, for all primes p. To see this, we note that tensoring over Z with Z_{(p)} preserves exactness, and hence

(4.3)  Ext_{ZΓ_{G}}^{n}(M, N) ⊗_{Z} Z_{(p)} ≅ Ext_{Z_{(p)}Γ_{G}}^{n}(M ⊗_{Z} Z_{(p)}, N ⊗_{Z} Z_{(p)}).

We also have the following:

Proposition 4.4. Let M and N be two ZΓ_{G}-modules, where M(H) is Z-torsion free for all H ∈ F. Then, for every n > l(M), there is an isomorphism

Ext_{ZΓ_{G}}^{n}(M, N) ≅ ⨁_{p||G|} Ext_{Z_{(p)}Γ_{G}}^{n}(M_{p}, N_{p})

where M_{p} = Z_{(p)} ⊗_{Z} M and N_{p} = Z_{(p)} ⊗_{Z} N.

Proof. From Proposition 4.2 we know that Ext_{ZΓ_{G}}^{n}(M, N) is a finite abelian group with exponent dividing some power of |G|, when n > l(M). Now the flatness of Z_{(p)} over Z implies as above that Ext_{ZΓ_{G}}^{n}(M, N) is the direct sum of its p-localizations, for all p | |G|. We then apply the isomorphism (4.3). □

To complete the proof of Proposition 3.13, we also need the following standard result in homological algebra (see [7, Chap. VI, 2.1] for the case of modules over rings):

Proposition 4.5. A right RΓ_{G}-module M admits a finite projective resolution if and only if there exists an integer ℓ_{0} ⩾ 0 such that Ext_{RΓ_{G}}^{n}(M, N) = 0, for all n > ℓ_{0} and all right RΓ_{G}-modules N.

Proof. If M admits a finite projective resolution of length k, then Ext_{RΓ_{G}}^{n}(M, N) = 0 for n > k and any RΓ_{G}-module N. Conversely, if Ext_{RΓ_{G}}^{n}(M, N) = 0 for n > ℓ_{0} and any N, then consider the kernel Z_{m} of the boundary map ∂_{m} : P_{m} → P_{m−1} in the projective resolution P of M. It follows that

Ext_{RΓ_{G}}^{1}(Z_{m}, N) ≅ Ext_{RΓ_{G}}^{m+2}(M, N) = 0

for any RΓ_{G}-module N, provided m+2 > ℓ_{0}, and so Z_{m} is projective if we take m = ℓ_{0}−1. This gives a finite projective resolution of length ℓ_{0} over RΓ_{G}. □

We now recall the definition of a Mackey functor (following Dress [13]). Let G be a finite group and D(G) denote the Dress category of finite G-sets and G-maps. A bivariant functor

M = (M^{∗}, M_{∗}): D(G) → R-Mod

consists of a contravariant functor

M^{∗}: D(G) → R-Mod

and a covariant functor

M_{∗}: D(G) → R-Mod.

The functors are assumed to coincide on objects. Therefore, we write M(S) = M_{∗}(S) = M^{∗}(S) for a finite G-set S. If f: S → T is a morphism, we often use the notation f_{∗} = M_{∗}(f) and f^{∗} = M^{∗}(f). If S = G/H and T = G/K with H ≤ K and f: G/H → G/K is given by f(eH) = eK, then we use the notation f_{∗} = Ind_{H}^{K} and f^{∗} = Res_{H}^{K}.

Definition 4.6 (Dress [13]). A bivariant functor is called a Mackey functor if it has the following properties:

(M1) For each pullback diagram

\xymatrix{X \ar[r]^{h} \ar[d]_{g} & Y \ar[d]^{k} \\ S \ar[r]_{f} & T}

of finite G-sets, we have h_{∗} ∘ g^{∗} = k^{∗} ∘ f_{∗}. (M2) The two embeddings S → S ⨆ T ⟵ T into the disjoint union define an isomor-

phism M^{∗}(S ⨆ T) ≅ M^{∗}(S) ⊕ M^{∗}(T).

Remark 4.7. There is a functor Or(G) → D(G) defined on objects by H ↦ G/H for every subgroup H ≤ G, and as the identity on morphism sets. By composition, any contravariant functor D(G) → R-Mod gives a right RΓ_{G}-module, with respect to any given family of subgroups F of G.

In the statement of Theorem 4.11 we will use the examples R[S^{?} ]: D(G) → R-Mod, defined in (3.4) for any finite G-set S.

The following example and lemma will be used in the proof of Theorem 5.9.

Example 4.8. Let Q ∈ F and let V be a right R[W_{G}(Q)]-module, where W_{G}(Q) = N_{G}(Q)/Q. Then we define a bivariant functor D_{Q}(V): D(G) → R-Mod on objects by setting

D_{Q}(V)(S) = Hom_{R[W_{G}(Q)]}(R[S^{Q}], V)

for any finite G-set S. For any G-map f: S → T we have a W_{G}(Q)-map f^{Q}: S^{Q} → T^{Q}, which induces a homomorphism

f^{∗}: Hom_{R[W_{G}(Q)]}(R[T^{Q}], V) → Hom_{R[W_{G}(Q)]}(R[S^{Q}], V)

by composition. To define the covariant map f_{∗}, let φ_{S}: R[S^{Q}] → V be an R[W_{G}(Q)]- homomorphism, and define f_{∗}(φ_{S}) = φ_{T} by

f_{∗}(φ_{S})(t) = φ_{T} (t) = ∑_{s∈S^{Q},f(s)=t} φ_{S}(s)

It is not hard to verify that D_{Q}(V) is actually a Mackey functor. The axiom (M1) follows because the Q-fixed sets in a pull-back diagram of G-sets give again a pull-back diagram. The axiom (M2) is immediate.

Definition 4.9. For any RΓ_{G}-module N, we define DN = ∑_{Q∈Iso(Γ_{G})} D_{Q}(N(Q)) and define j: N → DN as the direct sum of the adjoints of id: N(Q) → N(Q), for each Q ∈ Iso(Γ_{G}). Let CN denote the cokernel of j. For k ≥ 0, define inductively C^{0}N = N and C^{k}N = C(C^{k−1}N), together with the induced maps C^{k} → DC^{k}.

Here is a dual construction to the E-resolution given in [20, 17.13].

Lemma 4.10. For any RΓ_{G}-module N, the finite length sequence

0 → N ⟶^{j} DN → DCN → ⋯ → DC^{m}N → 0

is an exact coresolution of Mackey functors, for some m ≥ 0.

Proof. For any RΓ_{G}-module N, the map j: N → DN defined above is injective, so we have a short exact sequence

0 → N ⟶^{j} DN → CN → 0.

Iterating the above process, we obtain

0 → CN → DCN → C^{2}N → 0

and so on. By splicing, we get an exact sequence, or coresolution:

0 → N ⟶^{j} DN → DCN → ⋯ → DC^{k−1}N → DC^{k}N → ⋯

When N is a RΓ_{G}-module of a finite length, which is the case in our situation, this coresolution has a finite length. To check this, we use the definition of D_{Q}(V) in Example 4.8 to get

D_{Q}(V)(K) = Hom_{R[W_{G}(Q)]}(R[(G/K)^{Q}], V)

for any R[W_{G}(Q)]-module V. Therefore D_{Q}(V)(K) is only nonzero for (Q) ≤ (K), and at (Q) = (K) the R[W_{G}(K)]-module D_{Q}(V)(K) is isomorphic to V, via the isomorphism W_{G}(Q) ≅ W_{G}(K) induced by conjugation. This shows that the length of the module C^{k}N is properly smaller than the length of C^{k−1}N for all k ≥ 1. □

We will prove Theorem 5.1 by showing that H ↦ Ext_{RΓ_{H}}^{∗}(M, N) has a cohomological Mackey functor structure which is conjugation invariant. First we describe the Mackey functor structure on Hom_{RΓ_{?}}(M, N).

Theorem 4.11. For a right RΓ_{G}-module M and a Mackey functor N, let

Hom_{RΓ_{?}}(M, N): D(G) → R-Mod

denote the function defined by S ↦ Hom_{RΓ_{G}}(M ⊗_{R} R[S^{?} ], N) for any finite G-set S. Then Hom_{RΓ_{?}}(M, N) inherits a Mackey functor structure.

Proof. We will first define the induction and restriction maps to see that Hom_{RΓ_{?}}(M, N) is a bifunctor. For f: S → T a G-map, the restriction map

f^{∗}: Hom_{RΓ_{G}}(M ⊗_{R} R[T ^{?} ], N) → Hom_{RΓ_{G}}(M ⊗_{R} R[S^{?} ], N)

is the composition with M ⊗_{R} R[S^{?} ] ⟶^{id⊗f̃} M ⊗_{R} R[T ^{?} ] where f̃ denotes is the linear extension of the map induced by f. Since the functors R[S^{?} ] satisfy axiom (M2), so does Hom_{RΓ_{?}}(M, N).

For f: S → T a G-map, we define the induction map

f_{∗}: Hom_{RΓ_{G}}(M ⊗_{R} R[S^{?} ], N) → Hom_{RΓ_{G}}(M ⊗_{R} R[T ^{?} ], N)

in the following way: let φ_{S}: M ⊗_{R} R[S^{?} ] → N be given. We will describe the homomorphism φ_{T} = f_{∗}(φ_{S}).

φ_{T} (V)(x ⊗ α) = F_{∗}(φ_{S}(U)(F^{∗}(x) ⊗ β))

for x ∈ M(V) and α: V → T, where U, β and F are given by the pull-back

\xymatrix{U \ar[r]^{β} \ar[d]_{F} & S \ar[d]^{f} \\ V \ar[r]_{α} & T}

It is easy to check that this formula for φ_{T} gives an RΓ_{G}-homomorphism, using the assumption that N is a Mackey functor.

We need to check axiom (M1) for Hom_{RΓ_{?}}(M, N). For a given pull-back square

\xymatrix{X \ar[r]^{h} \ar[d]_{g} & Y \ar[d]^{k} \\ S \ar[r]_{f} & T}

we need to show that h_{∗} ∘ g^{∗} = k^{∗} ∘ f_{∗}. Let γ: V → Y be any G-map, and consider the extended pull-back diagram

\xymatrix{U \ar[r]^{δ} \ar[d]_{F} & X \ar[r]^{g} \ar[d]_{h} & S \ar[d]^{f} \\ V \ar[r]_{γ} & Y \ar[r]_{k} & T}

The maps α = k ∘ γ and β = g ∘ δ may be used to compute f_{∗}(φ_{S}) as above, and the left-hand square may be used to compute h_{∗}.

For any element φ_{S}: M ⊗_{R} R[S^{?} ] → N, we have

(k^{∗} ∘ f_{∗}(φ_{S}))(V)(x ⊗ γ) = (f_{∗}(φ_{S}) ∘ (id ⊗ k))(V)(x ⊗ γ)
= f_{∗}(φ_{S})(V)(x ⊗ (k ∘ γ))
= F_{∗}(φ_{S}(U)(F^{∗}(x) ⊗ (g ∘ δ))

for any x ∈ M(V) and γ: V → Y. On the other hand,

(h_{∗} ∘ g^{∗}(φ_{S}))(V)(x ⊗ γ) = F_{∗}((g^{∗}φ_{S})(U)(F^{∗}(x) ⊗ δ))
= F_{∗}(φ_{S}(U)(F^{∗}(x) ⊗ (g ∘ δ))

for any x ∈ M(V) and γ: V → Y, so the formula (M1) is verified. □

As an immediate consequence, for any subgroup H ≤ K the G-map f: G/H → G/K induces a restriction map

Res_{H}^{K}: Hom_{RΓ_{K}}(M, N) → Hom_{RΓ_{H}}(M, N)

defined as the composition of the map

f^{∗}: Hom_{RΓ_{G}}(M ⊗_{R} R[G/K^{?} ], N) → Hom_{RΓ_{G}}(M ⊗_{R} R[G/H^{?} ], N)

with the ‘Shapiro’ isomorphisms:

Hom_{RΓ_{G}}(M ⊗_{R} R[G/H^{?} ], N) ≅ Hom_{RΓ_{H}}(M, N)

and

Hom_{RΓ_{G}}(M ⊗_{R} R[G/K^{?} ], N) ≅ Hom_{RΓ_{K}}(M, N)

given by [35, Cor. 2.12] and the adjointness property (compare [2, Lemma 2.8.4]). Similarly, we have the induction map

Ind_{H}^{K}: Hom_{RΓ_{H}}(M, N) → Hom_{RΓ_{K}}(M, N)

defined by composing the Shapiro isomorphisms with f_{∗}.

Remark 4.12. Since Res_{H}^{G} preserves projectives, we see that P ⊗_{R} R[G/H^{?} ] is projective over RΓ_{G} whenever P is projective over RΓ_{G} (check the categorical lifting property directly or apply Lemma 3.2).

Proposition 4.13. Let C be a chain complex of right RΓ_{G}-modules and N be a Mackey functor. Then, the cochain complex

C^{∗} = Hom_{RΓ_{?}}(C, N)

with the differential δ: Hom_{RΓ_{?}}(C_{i}, N) → Hom_{RΓ_{?}}(C_{i+1}, N) given by δ(φ) = φ ∘ ∂ is a cochain complex of Mackey functors.

Proof. We have seen that each C^{i} = Hom_{RΓ_{?}}(C_{i}, N) is a Mackey functor by Theorem 4.11. We just need to show that the coboundary maps are Mackey functor maps. Given f: S → T we need to show the following diagram commutes:

\xymatrix{Hom_{RΓ_{G}}(C_{i} ⊗ R[S^{?} ], N) \ar[r]^{δ_{S}} \ar[d]^{f_{∗}} & Hom_{RΓ_{G}}(C_{i+1} ⊗ R[S^{?} ], N) \ar[d]^{f_{∗}} \\ Hom_{RΓ_{G}}(C_{i} ⊗ R[T ^{?} ], N) \ar[u]^{f^{∗}} \ar[r]^{δ_{T}} & Hom_{RΓ_{G}}(C_{i+1} ⊗ R[T ^{?} ], N) \ar[u]^{f^{∗}}}

The proof of commutativity for f^{∗} is easy. In this case, it follows from the commutativity of the following diagram:

\xymatrix{C_{i} ⊗ R[S^{?} ] \ar[d]^{id⊗f} & C_{i+1} ⊗ R[S^{?} ] \ar[l]_{∂⊗id} \ar[d]^{id⊗f} \\ C_{i} ⊗ R[T ^{?} ] & C_{i+1} ⊗ R[T ^{?} ] \ar[l]_{∂⊗id}}

For f_{∗} we check the commutativity directly: let φ_{S}: C_{i} ⊗ R[S^{?} ] → N be an RΓ_{G}-map. For x ∈ C_{i+1}(V) and α: V → T, we have

[(δ_{T} ∘ f_{∗})φ_{S}](x ⊗ α) = (f_{∗}φ_{S})(∂x ⊗ α)
= F_{∗}[φ_{S}(F^{∗}(∂x) ⊗ β)]

where

\xymatrix{U \ar[r]^{β} \ar[d]_{F} & S \ar[d]^{f} \\ V \ar[r]_{α} & T}

on the other hand,

[(f_{∗} ∘ δ_{S})φ_{S}](x ⊗ α) = F_{∗}[(δ_{S}φ_{S})(F^{∗}(x) ⊗ β)]
= F_{∗}[φ_{S} ∘ (∂ ⊗ id)(F^{∗}(x) ⊗ β)]
= F_{∗}[φ_{S}(∂F^{∗}(x) ⊗ β)]

since ∂F^{∗} = F^{∗}∂, we are done. □

Corollary 4.14. Let M be an RΓ_{G}-module and N be a Mackey functor. Then,

Ext_{RΓ_{?}}^{∗}(M, N)

has a Mackey functor structure. As a Mackey functor Ext_{RΓ_{?}}^{∗}(M, N) is equal to the homology of the cochain complex of Mackey functors Hom_{RΓ_{?}}(P, N) where P is a projective resolution of M as an RΓ_{G}-module.

Proof. To compute the Ext-groups, note that S ↦ P ⊗_{R} R[S^{?} ] is a projective resolution of the module S ↦ M ⊗_{R} R[S^{?} ], for every finite G-set S. □

Remark 4.15. It follows that a version of the Eckmann-Shapiro isomorphism

Ext_{RΓ_{G}}^{∗}(M ⊗ R[G/H^{?} ], N) ≅ Ext_{RΓ_{H}}^{∗}(Res_{H}^{G} M, Res_{H}^{G} N)

holds for the Ext-groups over the orbit category (compare [2, 2.8.4]).

Remark 4.16. If N is a Green module over a Green ring G, then the Mackey functor Ext_{RΓ_{?}}^{∗}(M, N) also inherits a Green module structure over G. The basic formula is a pairing

G(S) × Hom_{RΓ_{?}}(M ⊗_{R} R[S^{?} ], N) → Hom_{RΓ_{?}}(M ⊗_{R} R[S^{?} ], N)

induced by the Green module pairing G × N → N. For any z ∈ G(S), x ∈ M(V), and α: V → S, we define

(z · φ_{S})(V)(x ⊗ α) = α^{∗}(z) · φ_{S}(V)(x ⊗ α)

for any φ_{S}(V): M(V)⊗_{R}R Mor(S, V) → N(V). The check that this pairing gives a Green module structure is left to the reader. □

5. The proof of Theorem C

The main purpose of this section to prove the following theorem.

Theorem 5.1. Let G be a finite group, R = Z_{(p)}, and F be a family of subgroups in G. Suppose H ≤ G controls p-fusion in G. Then,

Res_{H}^{G}: Ext_{RΓ_{G}}^{n}(M, N) → Ext_{RΓ_{H}}^{n}(Res_{H}^{G} M, Res_{H}^{G} N)

is an isomorphism for n ⩾ 0, provided that M is an RΓ_{G}-module and N is a cohomological Mackey functor satisfying the condition that C_{G}(Q) acts trivially on N(Q) and M(Q) for all p-subgroups Q ≤ H, with Q ∈ F.

Certain Mackey functors (called cohomological) are computable by restriction to the p-Sylow subgroups and the conjugation action of G (see [7, Chap. XII, §10], [19]).

If H ≤ G is a subgroup, and n ∈ N_{G}(H) then the G-map f: G/H → G/H defined by f(eH) = nH has an associated conjugation homomorphism c_{n}(h) = n^{−1}hn ∈ H, for all h ∈ H. For an arbitrary RΓ_{G}-module M, the induced maps f^{∗} need not be the identity on M(G/H) even if c_{n} = id (e.g. if n ∈ C_{G}(H)).

Definition 5.2. We say a Mackey functor is cohomological (over F) if

Ind_{H}^{K} Res_{H}^{K}(u) = |K: H| · u

for all u ∈ M(K), and all H ≤ K (for all K ∈ F). An RΓ_{G}-module M with respect to a family F is called conjugation invariant if C_{G}(Q) acts trivially on M(Q) for all Q ∈ F. A Mackey functor is called conjugation invariant if it is conjugation invariant as a functor over the corresponding orbit category.

The following lemma will be used in the proof of Theorem 5.9.

Lemma 5.3. Let Q ∈ F and let V be a right R[W_{Q}(Q)]-module. If F is a family of psubgroups, and R = 𝔽_{p}, then D_{Q}(V): D(G) → R-Mod is a cohomological Mackey functor over F. If C_{G}(Q) acts trivially on V, then D_{Q}(V) is conjugation invariant.

Proof. Since all subgroups in F are p-groups, for the first part we only need to show that the composite Ind_{H}^{K} Res_{H}^{K}(u) = p · u, for K ∈ F and H ≤ K a normal of index p.

Let f: G/H → G/K be the G-map given by gH ↦ gK. Consider the induced map f^{Q}: (G/H)^{Q} → (G/K)^{Q}. Take t ∈ (G/K)^{Q}. If there is no s ∈ (G/H)^{Q} such that f(s) = t, then the transfer is trivially zero. Suppose that there is at least one element s = gH which is fixed by Q and maps to t = gK. Let k_{1}, …, k_{p} be coset representatives of H in K. Since k_{i} normalizes H, the element gk_{i}H ∈ (G/H)^{Q} for each i. Therefore, there are exactly p different s ∈ (G/H)^{Q} that map to t. It follows that f_{∗} ∘ f^{∗} is multiplication by p, as required. Since we are working here over the finite field 𝔽_{p}, all the composites f_{∗} ∘ f^{∗} = 0.

We now show that D_{Q}(V) is conjugation invariant if C_{G}(Q) acts trivially on V. In other words, we claim that for all K ∈ F, the centralizer C_{G}(K) acts trivially on Hom_{R]W_{G}(Q)]}(R[G/K]^{Q}, V). Consider the way the action is defined: let c ∈ C_{G}(K) and φ: R[G/K]^{Q} → V be an R[W_{G}(Q)]-map. Then (cφ)(gK) = φ(gcK). On the other hand since gK ∈ (G/K)^{Q}, we have Q^{g} ≤ K. So, c centralizes Q^{g}. This means gcg^{−1} centralizes Q and hence acts trivially on V. This gives

φ(gcK) = φ(gcg^{−1}gK) = gcg^{−1}φ(gK) = φ(gK)

Therefore (cφ)(gK) = φ(gK) for all gK. This shows that c ∈ C_{G}(K) acts as the identity on Hom_{R[W_{G}(Q)]}(R[G/K]^{Q}, V). □

The cohomological and conjugation properties are inherited by the Ext-functors.

Proposition 5.4. Let M and N be RΓ_{G}-modules relative to some family F.

(i) If N is a cohomological Mackey functor over F, then Ext_{RΓ_{?}}^{∗}(M, N) is a cohomological Mackey functor over all subgroups H ≤ G.

(ii) If both M and N are conjugation invariant with respect to F, then Ext_{RΓ_{?}}^{∗}(M, N) is conjugation invariant with respect to all subgroups H ≤ G.

Proof. We have seen that for f: S → T, the induced maps

\xymatrix{Hom_{RΓ_{G}}(M ⊗ R[S^{?} ], N) \ar[r]^{f_{∗}} & Hom_{RΓ_{G}}(M ⊗ R[T ^{?} ], N) \ar[l]^{f^{∗}}}

satisfy the property that

[(f_{∗} ∘ f^{∗})φ_{T}](V)(x ⊗ α) = F_{∗}[f^{∗}(φ_{T})(U)(F^{∗}(x) ⊗ β)]
= F_{∗}[φ_{T} (U)(F^{∗}(x) ⊗ (f ∘ β))]
= F_{∗}[φ_{T} (U)(F^{∗}(x) ⊗ (α ∘ F))]
= (F_{∗} ∘ F^{∗})[φ_{T} (V)(x ⊗ α)]

for all x ∈ M(V) and α: V → T. In the last equality we used the invariance of φ_{T} with respect to the G-map F: U → V (our notation comes from the definition of f_{∗} above). Hence, if f: G/H → G/K and F_{∗} ∘ F^{∗} is multiplication by |K: H| (this follows from a count of double cosets), then f_{∗} ∘ f^{∗} is also multiplication by |K: H|.

Let M and N be conjugation invariant right RΓ_{G}-modules, and let P be a projective resolution of M over RΓ_{G}. To show that Ext_{RΓ_{?}}^{∗}(M, N) is conjugation invariant, it is enough to show that the chain map induced by the conjugation action on Hom_{RΓ_{?}}(P, N) is homotopy equivalent to the identity. We remark that the action of an element c ∈ C_{G}(H) gives an automorphism J_{c}: Or_{F} H → Or_{F} H, and induces an RΓ_{H}-module chain map P(J_{c}): Res_{H}^{G}(P) → Res_{H}^{G}(P).

If f: G/H → G/H is given by eH ↦ cH where c ∈ C_{G}(H), then for each degree i,

f_{i}^{∗}: Hom_{RΓ_{G}}(P_{i} ⊗ R[G/H^{?} ], N) → Hom_{RΓ_{G}}(P_{i} ⊗ R[G/H^{?} ], N)

is given by

f_{i}^{∗}(φ_{S})(U)(x ⊗ α) = φ_{S}(U)(x ⊗ f ∘ α)

where S = G/H, x ∈ P_{i}(U), and α: U → G/H is a G-map. In other words, f_{i}^{∗} = Hom_{RΓ_{G}}(λ_{i}, id), where λ_{i}(x ⊗ α) = x ⊗ f ∘ α defines a chain map

λ: P ⊗ R[G/H^{?} ] → P ⊗ R[G/H^{?} ].

We may assume that U = G/K with K ∈ F. Let α(eK) = gH. The conjugation action of c ∈ C_{G}(H) on M(U) or N(U) is given by the G-map F: G/K → G/K, where F (eK) = gcg^{−1}K and f ∘ α = α ∘ F. We remark that z := gcg^{−1} ∈ C_{G}(K), since K ⊆ gHg^{−1}, and that P^{∗}(F) = P(J_{z})(K). Notice that

f_{i}^{∗}(φ_{S})(U)(x ⊗ α) = (φ_{S}(U)(x · P_{i}^{∗}(F)^{−1} ⊗ α)) · N^{∗}(F),

showing that the maps f_{i}^{∗} are just given by the natural action maps of c on the domain and range of the Hom. Now observe that

P(J_{z}): Res_{K}^{G}(P) → Res_{K}^{G}(P)

is a chain map lifting M(J_{z}): Res_{K}^{G}(M) → Res_{K}^{G}(M). Since M is conjugation invariant, it follows that P(J_{z}) ≃ id by uniqueness (up to chain homotopy) of lifting in projective resolutions. Therefore λ_{1} := λ ∘ (P^{∗}(F) ⊗ id) ≃ λ, and f^{∗} ≃ Hom(λ_{1}, id). But for all x ∈ P_{i}(U), we have

Hom(λ_{1}, id)(φ_{S})(U)(x ⊗ α) = φ_{S}(U)(x · P_{i}^{∗}(F) ⊗ f ∘ α) = (φ_{S}(U)(x ⊗ α)) · N^{∗}(F),

and hence f^{∗}(φ_{S}) ≃ φ_{S}, by the conjugation invariance of N. □

Definition 5.5. For any subgroup H ≤ G, and any RΓ_{G}-modules M and N, an element α ∈ Ext_{RΓ_{H}}^{n}(M, N) is called stable with respect to G provided that

Res_{H∩^{g}H}^{H}(α) = Res_{H∩^{g}H}^{^{g}H} c_{H}^{g}(α)

for any g ∈ G. The map c_{H}^{g} is the induced map f_{∗} where f: G/H → G/^{g}H is the G-map given by xH → xg^{−1}(gHg^{−1}).

Theorem 5.6. Let R = Z_{(p)} and G be a finite group. For a right RΓ_{G}-module M and a cohomological Mackey functor N: D(G) → R-Mod, the restriction map

Res_{P}^{G}: Ext_{RΓ_{G}}^{n}(M, N) → Ext_{RΓ_{P}}^{n} (M, N)

is an isomorphism for n ⩾ 0 onto the stable elements, for any p-Sylow subgroup P ≤ G.

Proof. By Proposition 5.4(i), Ext_{RΓ_{?}}^{n}(M, N) is a cohomological Mackey functor. Now the result follows (as in [33, 2.2]) from the stable element method of Cartan and Eilenberg [7, Chap. XII, 10.1]. □

Remark 5.7. Since Ext_{RΓ_{?}}^{∗}(M, N) is a cohomological Mackey functor, it is a Green module over the trivial module \underline{R}, considered as a Green ring by defining Ind_{H}^{K}: \underline{R}(G/H) → \underline{R}(G/K) to be multiplication by |K : H| (see [19, Ex. 2.9]). It follows that Ext_{RΓ_{?}}^{∗}(M, N) is computable in the sense of Dress in terms of the p-Sylow subgroups (see [16, Ex. 5.9]).

The proof of Theorem 5.1. Let R = Z_{(p)} and G be a finite group. Let H ≤ G be a subgroup which controls p-fusion in G. For any cohomological Mackey functor F, the restriction map Res_{P}^{G} maps surjectively to the stable elements in F (P), for any p-Sylow subgroup P ≤ G. If H controls p-fusion in G, and F is conjugation invariant, then all elements in F (H) are stable and

Res_{H}^{G}: F (G) ⟶^{≈} F (H)

is an isomorphism. This follows by a standard argument used to prove one direction of Mislin’s theorem in group cohomology (see, for example, Symonds [33, Theorem 3.5] or Benson [2, Proposition 3.8.4]). We apply Proposition 5.4 and this remark to the cohomological Mackey functor F = Ext_{RΓ_{?}}^{n}(M, N), and the proof is complete. □

In the next section we will need a variation of this result.

Definition 5.8. We say the N is an atomic right RΓ_{G}-module of type Q ∈ F, if N = I_{Q}(N(Q)) where I_{Q} is the inclusion functor introduced in Section 2.

Theorem 5.9. Let G be a finite group, R = Z_{(p)}, and let F be a family of p-subgroups in G. Suppose H ≤ G controls p-fusion in G. Then, for RΓ_{G}-modules M and N,

Res_{H}^{G}: Ext_{RΓ_{G}}^{n}(M, N) → Ext_{RΓ_{H}}^{n}(Res_{H}^{G} M, Res_{H}^{G} N)

is an isomorphism for n ⩾ 0, provided that C_{G}(Q) acts trivially on M(Q) and N(Q) for all Q ∈ F.

Proof. Without loss of generality, we can assume that N is an atomic RΓ_{G}-module of type Q, with trivial C_{G}(Q)-action on N(Q). This follows from the 5-lemma (using the filtration of N in [20, 16.8]).

Furthermore, we may also assume that N(Q) is R-torsion free. To see this, observe that as an N_{G}(Q)/QC_{G}(Q)-module, N(Q) fits into a short exact sequence 0 → L → F → N(Q) → 0, where F is a free N_{G}(Q)/QC_{G}(Q)-module. By taking inflations of these modules, we can consider the sequence as a sequence of N_{G}(Q)/Q -modules and apply the functor I_{Q}. This shows that N fits into a sequence 0 → N′′ → N′ → N → 0, where both N′ and N′′ are conjugation invariant and atomic, with an R-torsion free module at Q.

Now let N_{p} = N ⊗ 𝔽_{p} = N/pN. By Lemma 4.10 we have a finite length coresolution

(5.10)  0 → N_{p} → DN_{p} → DCN_{p} → ⋯ → DC^{m}N_{p} → 0

for some m ≥ 0. Since F is family of p-groups, Lemma 5.3 shows that the Mackey functors DC^{i}N_{p} are cohomological over F and conjugation invariant, for 0 ≤ i ≤ m.

We can apply the functors Ext_{RΓ_{?}}^{∗}(M, −) to the coresolution (5.10). By Proposition 5.4, the Mackey functors Ext_{RΓ_{?}}^{∗}(M, DC^{i}N_{p}) are also cohomological and conjugation invariant. Therefore

Res_{H}^{G}: Ext_{RΓ_{G}}^{∗}(M, N_{p}) → Ext_{RΓ_{H}}^{∗}(M, N_{p})

is an isomorphism by Theorem 5.1 and the 5-lemma (using the coresolution). Furthermore, since N(Q) is R-torsion free, we have a short exact sequence

0 → N/p^{k−1} → N/p^{k} → N/p → 0,

for every k ≥ 1, and hence by “dévissage” we conclude that

(5.11)  Res_{H}^{G}: Ext_{RΓ_{G}}^{∗}(M, N/p^{k}) ⟶^{≈} Ext_{RΓ_{H}}^{∗}(M, N/p^{k})

is an isomorphism, for every k ≥ 1. To finish the proof it is enough to show that

Res_{H}^{G}: Ext_{RΓ_{G}}^{∗}(M, N) ⊗ Ẑ_{p} → Ext_{RΓ_{H}}^{∗}(M, N) ⊗ Ẑ_{p}

is an isomorphism. However, for P any projective resolution of M over RΓ_{G}, the complex

Hom_{RΓ_{G}}(P, N/p^{k}) = Hom_{RΓ_{G}}(P, N) ⊗ Z/p^{k}

is a cochain complex of finitely-generated R-modules. By the universal coefficient theorem in cohomology [30, p. 246], we have an exact sequence

0 → Ext_{RΓ_{G}}^{n}(M, N) ⊗ Z/p^{k} → Ext_{RΓ_{G}}^{n}(M, N/p^{k}) → Tor_{1}^{R}(Ext_{RΓ_{G}}^{n+1}(M, N), Z/p^{k}) → 0.

Since Ẑ_{p} = \underleftarrow{lim} Z/p^{k} and the inverse limit functor is left exact, we obtain an exact sequence

0 → Ext_{RΓ_{G}}^{n}(M, N) ⊗ Ẑ_{p} → \underleftarrow{lim} Ext_{RΓ_{G}}^{n}(M, N/p^{k}) → \underleftarrow{lim} Tor_{1}^{R}(Ext_{RΓ_{G}}^{n+1}(M, N), Z/p^{k}).

Now we compare this sequence via Res_{H}^{G} to the corresponding sequence for the subgroup H, and use the dévissage isomorphisms (5.11) on the middle term. This shows immediately that Res_{H}^{G} is injective on the first term, for all n ≥ 0. Since the functor Tor_{1}^{R} is left exact, we get Res_{H}^{G} injective on the third term as well. But now a diagram chase shows that Res_{H}^{G} is surjective on the first term. □

6. Chain complexes over orbit categories

In this section, we prove some theorems about chain complexes over orbit categories. In particular, Proposition 6.8, Proposition 6.4, and Theorem 6.7 will be used in the proof of Theorem A (see Section 9). Most of the results follow from Dold’s theory of algebraic Postnikov systems [11].

As before, G denote a finite group and F denote a family of subgroups of G. Throughout this section Γ_{G} = Or_{F} G and R is a commutative ring. For chain complexes C and D, the notation C ≃ D always means C is chain homotopy equivalent to D. For chain isomorphism the standard notation is C ≅ D. When we say C is a projective chain complex, we mean it is a chain complex of projective modules (which also means that it is projective in the category of chain complexes). A chain complex C is positive if C_{i} = 0 for i < 0.

We say that a chain complex C over RΓ_{G} has finite homological dimension (or hdim C is finite) if C is positive, and there exists an integer n such that H_{i}(C) = 0 for i > n. A chain complex C is finite if C is positive, and there exists an integer n such that C_{i} = 0 for i > n. We start with a well known observation about chain complexes.

Lemma 6.1. Let C be a projective chain complex of RΓ_{G}-modules which has finite homological dimension. Then, C is homotopy equivalent to a finite projective chain complex if and only if there is an integer n such that

Ext_{RΓ_{G}}^{i}(C, M) = 0 for i > n,

for all RΓ_{G}-modules M.

Proof. See Cartan-Eilenberg [7, Chap. XVII, 1.4] for chain complexes over rings. A similar argument as in Proposition 4.5 gives the result over the orbit category. □

Proposition 6.2. Let C be a projective chain complex of ZΓ_{G}-modules which has a finite homological dimension. Suppose that Z_{(p)} ⊗_{Z} C is chain homotopy equivalent to a finite projective chain complex for all p | |G|. Then, C is chain homotopy equivalent to a finite projective complex.

Proof. Let M be an RΓ_{G}-module. Consider the hypercohomology spectral sequence (see [3, 3.4.3]):

E_{2}^{s,t} = Ext_{ZΓ_{G}}^{s}(H_{t}(C), M)

which converges to Ext_{ZΓ_{G}}^{∗}(C, M). Since C has finite homological dimension, for all i > (l(Γ_{G}) + hdim C), the group Ext_{Z}^{i}(C, M) is a finite abelian group with exponent dividing a power of |G|. Here l(Γ_{G}) is the length of the orbit category, as defined in §2D, and hdim C denotes the largest integer n such that H_{n}(C) ≠ 0.

In particular, there is an integer k, independent from M, such that

Ext_{ZΓ_{G}}^{i}(C, M) ≅ ⨁_{p||G|} Ext_{Z_{(p)}Γ_{G}}^{i}(Z_{(p)} ⊗_{Z} C, M_{p})

for all i > k. Here M_{p} = Z_{(p)} ⊗_{Z} M. Now, since Z_{(p)} ⊗_{Z} C is homotopy equivalent to a finite projective complex for all p | |G|, there is an n such that

Ext_{ZΓ_{G}}^{i}(C, M) = 0

for all i > n and for all M. The result follows from the previous lemma. □

A chain complex version of Rim’s theorem also holds.

Proposition 6.3. Let R = Z_{(p)} and C be a projective chain complex over RΓ_{G} with finite homological dimension. Assume that F is a family of p-subgroups. Then, C is homotopy equivalent to a finite projective complex if and only if Res_{P}^{G} C is homotopy equivalent to a finite projective complex for any p-Sylow subgroup P of G.

Proof. One direction is clear (and holds without assumption on the family F). Conversely, suppose that Res_{P}^{G} C is homotopy equivalent to a projective complex with hdim = l. Let n be an integer bigger than both l and hdim C. Consider

\xymatrix{⋯ \ar[r] & Res_{P}^{G} C_{n+1} \ar[r] & Res_{P}^{G} C_{n} \ar[r]^{∂_{n}} & Res_{P}^{G} C_{n−1} \ar[r] & ⋯ \ar[r] & Res_{P}^{G} C_{0} \ar[r] & 0} .

We have

Ext_{RΓ_{P}}^{1} (Res_{P}^{G} im(∂_{n}), M) ≅ Ext_{RΓ_{P}}^{n+1} (Res_{P}^{G} C, M) = 0,

for every RΓ_{P} -module M. This gives that Res_{P}^{G} im(∂_{n}) is projective. By Theorem 3.9, we obtain that im(∂_{n}) has finite projective resolution. Thus, C is chain homotopy equivalent to a finite projective complex. □

We also prove a chain complex version of Theorem 5.9. Recall the definition of conjugation invariant RΓ_{G}-modules given in (5.2).

Proposition 6.4. Let G be a finite group, and F be a family of p-subgroups in G. Suppose H ≤ G controls p-fusion in G and R = Z_{(p)}. Let C^{H} be a positive projective chain complex of RΓ_{H}-modules such that the homology groups H_{i}(C^{H}) are conjugation invariant right RΓ_{H}-modules, for every i ⩾ 0. Then, the following holds:

(i) There exists a positive projective chain complex C^{G} of RΓ_{G}-modules such that Res_{H}^{G} C^{G} is homotopy equivalent to C^{H}.

(ii) If C^{H} is homotopy equivalent to a finite projective complex, then C^{G} is also homotopy equivalent to a finite complex.

For the proof we will need the theory of algebraic Postnikov systems due to Dold [11, §7]. According to this theory, given a positive projective chain complex C, there is a sequence of positive projective chain complexes C(i) indexed by positive integers such that f: C → C(i) induces a homology isomorphism for dimensions ≤ i. Moreover, there is a tower of maps

\xymatrix{ & C(i) \ar[d] \\  & C(i − 1) \ar@{-->}[d] \ar[r]^{α_{i}} & Σ^{i+1}P(H_{i}) \\ C \ar[dr] \ar[ur] \ar[uur] \ar[r] & C(1) \ar[d] \ar[r]^{α_{2}} & Σ^{3}P(H_{2}) \\  & C(0) \ar[r]^{α_{1}} & Σ^{2}P(H_{1})}

such that C(i) = Σ^{−1}C(α_{i}), where C(α_{i}) denotes the algebraic mapping cone of α_{i}, and P(H_{i}) denotes a projective resolution of the homology module H_{i}.

Recall that the algebraic mapping cone of a chain map f: C → D is defined as the chain complex C(f) = D ⊕ ΣC with boundary map given by ∂(x, y) = (∂x + f(x), ∂y).

Note that Σ^{n} is the shift operator for chain complexes which is defined by (Σ^{n}C)_{i} = C_{i−n} for every integer n.

The algebraic Postnikov system has similar properties to the Postnikov system in homotopy theory. The maps α_{i}: C(i − 1) → Σ^{i+1}P(H_{i}) are called k-invariants and they are well defined up to chain homotopy equivalence. We can consider the k-invariants as classes in Ext_{RΓ_{G}}^{i+1}(C(i − 1), H_{i}), since there is an isomorphism

[C(i − 1), Σ^{i+1}P(H_{i})] ≅ Ext_{RΓ_{G}}^{i+1}(C(i − 1), H_{i})

between chain homotopy classes of chain maps and the Ext-groups of chain complexes (see Dold [11] for details). The k-invariants α_{i} ∈ Ext_{RΓ_{G}}^{i+1}(C(i − 1), H_{i}) are defined inductively and they uniquely specify C up to chain homotopy equivalence.

We also need a lifting result for RΓ_{H}-modules.

Lemma 6.5. Let G be a finite group, and F be a family of p-subgroups in G. Suppose H ≤ G controls p-fusion in G. Then the restriction map M ↦ Res_{H}^{G}(M) gives a bijection between the isomorphism classes of conjugation invariant right RΓ_{G}-modules and conjugation invariant right RΓ_{H}-modules.

Proof. A conjugation invariant right RΓ_{G}-module M is a functor Or_{F} G → R-Mod which factors through the quotient category Or_{F} G → Sub_{F} G. Here Sub_{F} G has objects K ∈ F and morphisms Mor_{Sub_{F} G}(K, L) = Mor_{Or_{F} G}(G/K, G/L)/C_{G}(K), where an element c ∈ C_{G}(K) acts on a G-map defined by f(eK) = gL via the composition eK ↦ cgL (see [21, p. 206]).

Consider the functor F: Or_{F} H → Or_{F} G given on objects by H/K ↦ G/K (see Section 3), and on morphisms by induced maps. First note that every object of Sub_{F} G is isomorphic to an object of Sub_{F} H, since every p-subgroup of G is conjugate to a subgroup of H. In addition, F induces a bijection of morphism sets

Mor_{Sub_{F} H}(K, L) → Mor_{Sub_{F} G}(K, L)

since H controls p-fusion in G. Suppose that F (f_{1}) ≈ F (f_{2}), where f_{1}(eK) = h_{1}L and f_{2}(eK) = h_{2}L, for some h_{1}, h_{2} ∈ H. By assumption, there exists c ∈ C_{G}(K) such that ch_{2}L = h_{1}L, or h_{1}^{−1}ch_{2} ∈ L ≤ H. But this implies c ∈ C_{H}(K) so f_{1} ≈ f_{2} and F is injective on morphisms. Given f: G/K → G/L with K ≤ H, f(eK) = gL and g^{−1}Kg ⊆ L ≤ H, we have g = ch for some c ∈ C_{G}(K) and h ∈ H, because H controls p-fusion in G. Hence f ≈ F (f_{1}), where f_{1}(eK) = hL and F is surjective on morphisms.

Therefore the functor F: Or_{F} H → Or_{F} G induces an equivalence of categories

F̄: Sub_{F} H ≈ Sub_{F} G

by [24, IV.4, Theorem 1, p. 91]. □

Proof of Proposition 6.4. Part (ii) follows from Proposition 6.3, so it is enough to prove the existence of C^{G}. By Lemma 6.5, for each i ⩾ 0 there exists a conjugation invariant right RΓ_{G}-module H_{i}^{G} such that Res_{H}^{G}(H_{i}^{G}) = H_{i}(C^{H}).

Consider the Postnikov tower for C^{H}. Since C^{H}(0) = P(H_{0}(C^{H})) there is a complex C^{G}(0) such that Res_{H}^{G} C^{G}(0) ≃ C^{H}(0). In this case, the complex C^{G}(0) can be taken as a projective resolution of H_{0}^{G}. Now, we will show that such a lifting exists for C^{H}(i) for all i. For this we prove a slightly stronger statement so that we can carry out an induction. We claim that the following holds for all n ⩾ 0.

(i) C^{H}(n) lifts to a chain complex C^{G}(n)

(ii) The restriction map

Res_{H}^{G}: Ext_{RΓ_{G}}^{∗}(C^{G}(n), N) → Ext_{RΓ_{H}}^{∗}(C^{H}(n), Res_{H}^{G} N)

is an isomorphism for all ∗ ⩾ 0 and for every RΓ_{G}-module N which is conjugation invariant.

We have already shown that C^{H}(0) lifts to C^{G}(0). For the second property, first observe that that C^{G}(0) is chain homotopy equivalent to a chain complex with single module H_{0}^{G} and similarly, C^{H}(0) ≃ H_{0}^{H}. So, we need to show that

Res_{H}^{G}: Ext_{RΓ_{G}}^{∗}(H_{0}^{G}, H_{1}^{G}) → Ext_{RΓ_{H}}^{∗}(H_{0}^{H}, H_{1}^{H})

is an isomorphism. This follows from Theorem 5.9, because of our assumption on homology groups.

Now, assume that both (i) and (ii) hold for n = i − 1. Then, take

α_{i}^{G} ∈ Ext_{RΓ_{G}}^{i+1}(C^{G}(i − 1), H_{i}^{G})

which corresponds to the class α_{i}^{H} ∈ Ext_{RΓ_{H}}^{i+1}(C^{H}(i−1), H_{i}^{H}) under the isomorphism given in (ii). Let C^{G}(i) = Σ^{−1}C(α_{i}^{G}). Then, we have a short exact sequence of chain complexes

\xymatrix{0 \ar[r] & C^{G}(i) \ar[r] & C^{G}(i − 1) \ar[r]^{α_{i}^{G}} & Σ^{i+1}P(H_{i}^{G}) \ar[r] & 0}

Since Res_{H}^{G} α_{i}^{G} = α_{i}^{H}, we have Res_{H}^{G} C^{G}(i) ≃ C^{H}(i). Now, we will show that (ii) holds for C^{G}(i). By the Five Lemma, it is enough to show that

Res_{H}^{G}: Ext_{RΓ_{G}}^{∗}(Σ^{i+1}P(H_{i}^{G}), N) → Ext_{RΓ_{H}}^{∗}(Σ^{i+1}P(H_{i}^{H}), N)

is an isomorphism for all ∗ ⩾ 0, and for every RΓ_{G}-module N which is conjugation invariant. But, this follows from Theorem 5.9. □

Now, we prove one of the main results of this section which allows us to glue p-local chain complexes. We first give a definition.

Definition 6.6. Let C be a chain complex over RΓ_{G}. We say that C has homology gaps of length n, if H_{i+k}(C) = 0 for 0 < k < n, whenever H_{i}(C) ≠ 0.

Theorem 6.7. Let G be a finite group of order m. For each prime p | m, let C^{(p)} be a positive projective chain complex of Z_{(p)}Γ_{G}-modules. Suppose that

(i) C^{(p)} has homology gaps of length > l(Γ_{G}), for all p | m.

(ii) there exists a graded ZΓ_{G}-module H such that H_{i}(C^{(p)}) ≅ Z_{(p)} ⊗_{Z} H_{i} for all i ≥ 0,

and for all p | m. Then, there is a projective chain complex C of ZΓ_{G}-modules such that Z_{(p)} ⊗_{Z} C ≃ C^{(p)}, for each prime p | m, and H_{i}(C) = H_{i} for i ≥ 0.

Proof. We will construct C inductively. The case i = 0 is trivial, because in this case we can take C(0) = P(H_{0}). Assume now that C(i − 1) has been constructed in such a way that Z_{(p)} ⊗_{Z} C(i − 1) ≃ C_{i−1}^{(p)} for all p | m. If H_{i} = 0, then we can take C(i) = C(i − 1) and it will satisfy the condition that Z_{(p)} ⊗_{Z} C(i) ≃ C_{i}^{(p)}. So, assume H_{i} is nonzero. If i + 1 > (l(Γ_{G}) + hdim C(i − 1)), then we have

Ext_{ZΓ_{G}}^{i+1}(C(i − 1), P(H_{i})) ≅ ⨁_{p|m} Ext_{Z_{(p)}Γ_{G}}^{i+1}(Z_{(p)} ⊗_{Z} C(i − 1), H_{i}^{(p)})
≅ ⨁_{p|m} Ext_{Z_{(p)}Γ_{G}}^{i+1}(C^{(p)}(i − 1), H_{i}^{(p)})

where H_{i}^{(p)} = Z_{(p)} ⊗_{Z} H_{i}. Note that the above condition on (i + 1) is satisfied since the distance between nonzero homology groups of C^{(p)} is bigger than l(Γ_{G}). Choose α_{i} ∈ Ext_{ZΓ_{G}}^{i+1}(C(i − 1), P(H_{i})) so that under the p-localization map, α_{i} is mapped to the i-th k-invariant α_{i}^{(p)} of the p-local complex C^{(p)}, for every p | m. Let C(i) = Σ^{−1}C(α_{i}). For each prime p | m, we have a diagram of the form

\xymatrix{0 \ar[r] & C(i) \ar[r] & C(i − 1) \ar[d]^{φ_{p}} \ar[r]^{α_{i}} & P(H_{i}) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & C^{(p)}(i) \ar[r] & C^{(p)}(i − 1) \ar[r]^{α_{i}^{(p)}} & P(H_{i}^{(p)}) \ar[r] & 0}

where the vertical map φ_{p} is given by the composition

φ_{p}: C(i − 1) → Z_{(p)} ⊗_{Z} C(i − 1) ≅ C^{(p)}(i − 1).

The first map in the above composition is induced by the usual inclusion of integers into p-local integers. From this diagram, it is clear that there is a map C(i) → C^{(p)}(i) which induces an isomorphism on homology when it is localized at p. Thus, it gives a chain homotopy equivalence Z_{(p)} ⊗_{Z} C(i) ≃ C^{(p)}(i), for p | m. This completes the proof. □

We conclude this section with a technique (used in the proof of Theorem A) for modifying the homology of a given (finite, projective) chain complex C over the orbit category. A projective resolution P → M has length ⩽ ℓ, provided that P_{i} = 0 for i > ℓ.

Proposition 6.8. Let Γ be an EI-category. Let φ: H_{k} → H′_{k} be an RΓ -module homomorphism, where H_{k} = H_{k}(C). Suppose that both kernel and cokernel of φ admit finite projective resolutions of length ⩽ ℓ, and that H_{k+j} = 0 for 1 ⩽ j < ℓ. Then there is a RΓ -chain complex C′ such that H_{i}(C′) = H_{i}(C), for i ≠ k, and H_{k}(C′) = H′_{k}.

Proof. First suppose that φ is surjective. Let

0 → P_{k+ℓ} → ⋯ → P_{k} → ker φ → 0

be a projective resolution for ker φ. Since C is exact in the range [k + 1, k + ℓ), we have a chain map

\xymatrix{⋯ \ar[r] & 0 \ar[r] \ar[d] & P_{k+ℓ} \ar[r] \ar[d]^{f_{k+ℓ}} & … \ar[r] & P_{k+1} \ar[r] \ar[d]^{f_{k+1}} & P_{k} \ar[r] \ar[d]^{f_{k}} & ker φ \ar[r] \ar@{^{(}->}[d] & 0 \\ ⋯ \ar[r] & C_{k+ℓ+1} \ar[r] & C_{k+ℓ} \ar[r] & … \ar[r] & C_{k+1} \ar[r] & Z_{k} \ar[r] & H_{k} \ar[r] & 0}

This gives a chain map f: P → C, where f_{k}: P_{k} → C_{k} is the composition of f_{k} with the inclusion Z_{k} ⊂ C_{k}. Let C′ = C(f) denote the mapping cone of f. The induced map

ker φ = H_{k}(P) → H_{k}(C) = H_{k}

on homology is given by the inclusion, and hence H_{k}(C′) = H′_{k}, with H_{i}(C′) = H_{i}(C) for i ≠ k.

Now suppose that φ is an injective map, so that

(6.9)  0 → H_{k} ⟶^{φ} H′_{k} → coker φ → 0

is exact. Let ϵ: P → coker φ be a projective resolution of coker φ of length ⩽ ℓ, indexed so that ϵ: P_{k} → coker φ → 0. We form the pull-back

[figure: 0; H_{k}; Ĥ_{k}; φ̂; P_{k}; ϵ; 0; 0; H_{k}; H′_{k}; coker φ; 0]

of the sequence (6.9) by ϵ, and note that Ĥ_{k} ≅ H_{k} ⊕ P_{k}. The chain complex

⋯ → C_{k+1} → C_{k} ⊕ P_{k} → C_{k−1} → ⋯ → C_{0} → 0

has homology Ĥ_{k} at i = k, and φ̂: Ĥ_{k} → H′_{k} is surjective. By the pull-back diagram,

ker φ̂ ≅ ker(ϵ: P_{k} → coker φ).

Since coker φ has a projective resolution of length ≤ ℓ, it follows that ker φ̂ has a projective resolution of length < ℓ. Hence the assumptions needed for the surjective case hold for φ̂: Ĥ_{k} → H′_{k}, and we are done by the argument above.

The general case is done by expressing the map φ: H_{k} → H′_{k} as the composition of a surjection and an injection. □

7. The finiteness obstruction

Let G be a finite group and F be a family of subgroups of G. The main result of this section is Theorem 7.6: given a finite projective chain complex C of ZΓ_{G}-modules, for Γ_{G} = Or_{F} G, we can obtain a finite free complex by taking join tensor of C with itself sufficiently many times. This result is an adaptation to the orbit category of the fundamental work of Swan [32]. We first introduce some definitions, based on the material in Lück [20, §10-11]).

Let Γ be an EI-category. We denote by K_{0}(ZΓ) the Grothendieck ring of isomorphism classes of projective ZΓ -modules and K_{0}(ZΓ, free) denote the Grothendieck ring of isomorphism classes of free ZΓ -modules (under direct sum M ⊕ N and tensor product M ⊗_{Z} N). We have an exact sequence of abelian groups

0 → K_{0}(ZΓ, free) → K_{0}(ZΓ) ⟶^{q} K̃_{0}(ZΓ) → 0

defining the quotient group K̃_{0}(ZΓ).

Note that K_{0}(ZΓ, free) is a subring, but not an ideal in general. This is because the tensor product of a free module with a projective module is not free in ZΓ. For example, if P is a projective module which is not free, then P ⊗ \underline{Z} ≅ P is not a free ZΓ_{G}-module although \underline{Z} is free when G ∈ F.

Given a finite projective chain complex of ZΓ -modules

C: 0 → C_{n} → C_{n−1} → ⋯ → C_{1} → C_{0} → 0

we define

σ(C) = ∑_{i=0}^{n}(−1)^{i}[C_{i}] ∈ K_{0}(ZΓ)

and

σ̃(C) = q(σ(C)) ∈ K̃_{0}(ZΓ).

The class σ̃(C) is called the finiteness obstruction since it is the only obstruction for C to be chain homotopy equivalent to a finite free chain complex.

From now on, we assume that all the chain complexes are positive and projective. As always, we assume all modules are finitely generated.

The following are standard results which show that σ̃(C) is an invariant, and that it is an obstruction for finiteness.

Lemma 7.1. If C and D are chain homotopy equivalent, then σ(C) = σ(D).

Proof. See [20, 11.2]. □

Lemma 7.2. Let C and D be finite chain complexes of projective ZΓ - modules. Then, σ(C ⊗_{Z} D) = σ(C) · σ(D).

Proof. See [20, 11.18] and the sharper result in [20, 11.24]. □

Lemma 7.3. Let C be a finite chain complex with σ̃(C) = 0. Then C is chain homotopy equivalent to a finite chain complex of free ZΓ -modules.

Proof. See Swan [32, Proposition 5.1]. □

Given two chain complexes of RΓ -modules C and D, consider the corresponding augmented complexes

C̃: ⋯ → C_{2} → C_{1} → C_{0} → \underline{R} → 0
D̃: ⋯ → D_{2} → D_{1} → D_{0} → \underline{R} → 0

Taking their tensor product, we obtain a complex of the form

C̃ ⊗_{R} D̃: ⋯ → C_{1} ⊕ D_{1} ⊕ C_{0} ⊗ D_{0} → C_{0} ⊕ D_{0} → \underline{R} → 0.

EQUIVARIANT CW-COMPLEXES AND THE ORBIT CATEGORY 33

Definition 7.4. We define the join tensor, denoted C ⋇ D, of two positive augmented chain complexes C and D by the formula

{ \atop C ⋇ D} = Σ (C̃ ⊗_{R} D̃) ,

where Σ denote the suspension of a chain complex defined by (ΣC)_{i} = C_{i−1} for all i. Lemma 7.5. Let C and D be finite chain complexes of projective ZΓ -modules. Then, σ(C ⋇ D) = σ(C) + σ(D) − σ(C) · σ(D). Proof. Note that (C ⋇ D)_{k} = C_{k} ⊕ D_{k} ⊕ ⨁_{i+j=k−1} C_{i} ⊗_{Z} D_{j}, for each k ≥ 0. Therefore,

σ(C ⋇ D) = ∑_{k}(−1)^{k}[C_{k}] + ∑_{k}(−1)^{k}[D_{k}] − ∑_{i+j=k−1}(−1)^{k}[C_{i} ⊗ D_{j}]

and the result follows. □

We often express the above formula by writing

(1 − σ(C ⋇ D)) = (1 − σ(C))(1 − σ(D)).

Whenever it is written in this way, one should understand it as a formal expression of the formula given in Lemma 7.5. The main theorem of this section is the following: Theorem 7.6. Let Γ_{G} = Or_{F} G where G is a finite group and F is a family of subgroups in G. Given a finite chain complex C of projective ZΓ_{G}-modules, there exists an integer n such that n-fold join tensor ⋇_{n}C of the complex C is chain equivalent to a finite complex of free ZΓ_{G}-modules.

We need to show that the finiteness obstruction σ̃(⋇_{n}C) vanishes for some n. In the proof we will use a result by Oliver and Segev [26]. Proposition 7.7. Let G be a finite group and let P and P′ be any two finitely generated projective ZG-modules. Then, P ⊗_{Z} P′ is stably free as a ZG-module. Proof. See [26, Proposition C.3]. □

We also need the following splitting theorem for K_{0}(ZΓ). Theorem 7.8. Let Γ be a EI-category. Then, the map

K_{0}(S) : K_{0}(ZΓ) → ⨁_{x∈Iso(Γ)} K_{0}(Z[x]),

defined by [P] → [S_{x}(P)] on each x ∈ Iso(Γ), is an isomorphism. The same holds when K_{0} is replaced by K̃_{0}. Proof. See Lück [10, Proposition 11.29]. □

As a consequence of this theorem, if Γ is finite then K̃_{0}(ZΓ) is finite: in this case Γ has finitely many isomorphism classes of objects x ∈ Ob(Γ), and Aut[x] is a finite group (apply Swan [31, Prop. 9.1]). In particular, if Γ = Or_{F} G, then the group K̃_{0}(Γ) is finite.

From now on we assume Γ_{G} = Or_{F} G for some finite group G, relative to some family F. The splitting theorem above can also be used to give a filtration of K̃_{0}(Γ_{G}). Recall that every projective ZΓ_{G}-module is of the form

P ≅ ⨁_{H∈T} E_{H}S_{H}P

where T is a set of representatives of conjugacy classes of elements in F. So, another way to express the above splitting theorem is to write

K_{0}(ZΓ_{G}) ≅ ⨁_{H∈T} K_{0}(ZΓ_{G})_{H}

where K_{0}(ZΓ_{G})_{H} = {[P] | E_{H}S_{H}P ≅ P }. Note that this is only a splitting as abelian groups, but using this we can give a filtration for the ring structure of K_{0}(ZΓ_{G}). Let

∅ = T_{0} ⊆ T_{1} ⊆ ⋯ ⊆ T_{m} = T

be a filtration of T such that if H ∈ T_{i} and K ∈ T_{j} and ^{g}H ≤ K for some g ∈ G, then i ⩽ j. This gives a filtration

0 = K_{0}(ZΓ_{G})_{0} ⊆ K_{0}(ZΓ_{G})_{1} ⊆ ⋯ ⊆ K_{0}(ZΓ_{G})_{m} = K_{0}(ZΓ_{G})

where

K_{0}(ZΓ_{G})_{i} = {[P] | P = ⨁_{H∈T_{i}} E_{H}S_{H}P }.

Lemma 7.9. Let V be a Z[N_{G}(H)/H]-module and W be a Z[N_{G}(K)/K]-module. Then,

E_{H}V ⊗_{Z} E_{K}W ≅ ⨁_{HgK∈H\G/K} E_{H∩^{g}K}(Res_{H∩^{g}K} E_{H}V ⊗_{Z} Res_{H∩^{g}K} E_{K}W).

Proof. Applying the definition, we get

E_{H}V ⊗_{Z} E_{K}W = (V ⊗ W) ⊗_{Z[Aut(G/H)×Aut(G/K)]} Z Map_{G}(?, G/H × G/K)

where Map_{G}(X, Y) denotes the set G-sets from X to Y (see [10, 11.30] for a similar computation). Since

G/H × G/K = ∐_{HgK∈H\G/K} G/(H ∩^{g}K),

the module E_{H}V ⊗_{R} E_{K}W decomposes as

⨁_{HgK∈H\G/K} E_{H∩^{g}K}U_{H∩^{g}K}

where U_{H∩^{g}K} are N_{G}(H ∩^{g}K)/(H ∩^{g}K)-modules. Applying S_{H∩^{g}K}, we find

U_{H∩^{g}K} = S_{H∩^{g}K}(E_{H}V ⊗_{Z} E_{K}W) = Res_{H∩^{g}K}(E_{H}V ⊗_{Z} E_{K}W)
= Res_{H∩^{g}K} E_{H}V ⊗_{Z} Res_{H∩^{g}K} E_{K}W.
□

Lemma 7.10. K_{0}(ZΓ_{G})_{i} is an ideal of K_{0}(ZΓ_{G}) .

EQUIVARIANT CW-COMPLEXES AND THE ORBIT CATEGORY 35

Proof. For E_{H}S_{H}P and E_{K}S_{K}Q, we have

E_{H}S_{H}P ⊗_{Z} E_{K}S_{K}Q = ⨁_{L} E_{L}V_{L}

where L = H ∩^{g}K for some g ∈ G. So, if H ∈ T_{i}, K ∈ T_{j}, and L ∈ T_{k}, then k ⩽ i, j. □

Now, Theorem 7.6 follows by induction from the following proposition. Proposition 7.11. Let C be a finite chain complex of projective ZΓ_{G}-modules. If σ̃(S_{H}C) = 0 for all H ∈ T ∖ T_{i}, then there is an n such that σ̃(S_{H}(⋇_{n}C)) = 0 for all H ∈ T ∖ T_{i−1}. Proof. An element in σ(C) can be expressed as a sum u + ∑_{j} v_{j} + w where

u = ∑_{H∈T_{i−1}} σ(E_{H}S_{H}C), ∑_{j} v_{j} = ∑_{H∈T_{i}∖T_{i−1}} σ(E_{H}S_{H}C), w = ∑_{H∈T ∖T_{i}} σ(E_{H}S_{H}C).

By Lemma 7.5, we have

1 − σ(⋇_{n}C) = (1 − σ(C))^{n} = (1 − (u + ∑_{j} v_{j} + w))^{n} ∈ K_{0}(ZΓ_{G})

So,

1 − σ(⋇_{n}C) ≡ (1 − (∑_{j} v_{j} + w))^{n} mod K_{0}(ZΓ_{G})_{i−1}

By Lemma 7.9, it is easy to see that v_{j} · v_{k} ≡ 0 (mod K_{0}(ZΓ_{G})_{i−1}), for j ≠ k. Note that

v_{j} · v_{j} ≡ stably free (mod K_{0}(ZΓ_{G})_{i−1})

by Proposition 7.7. To complete the proof, observe that modulo K_{0}(ZΓ_{G})_{i−1},

1 − σ(⋇_{n}C) ≡ (1 − (∑_{j} v_{j} + w))^{n}
≡ 1 + ∑_{k=1}^{n} \binom{n}{k}(−1)^{k}(∑_{j} v_{j} + w)^{k}
= 1 + ∑_{k=1}^{n} \binom{n}{k}(−1)^{k} ∑_{j} kv_{j}w^{k−1} + stably free
= 1 + n ∑_{k=1}^{n} ∑_{j} \binom{n − 1}{k − 1}(−1)^{k}v_{j}w^{k−1} + stably free.

This shows that σ(⋇_{n}C) is stably free for some n, since K̃_{0}(ZΓ_{G}) is a finite group. □

8. Realization of free chain complexes

Let X be G-CW complex, and let F be a family of subgroups of G. Throughout this section, R denotes a commutative ring and Γ_{G} denotes the orbit category Or_{F} G.

Definition 8.1. We say that a G-CW complex X has isotropy in F, provided that X^{H} ≠ ∅ implies H ∈ F, for all H ≤ G.

The main result of this section is Theorem 8.10, which shows that under certain conditions a finite free chain complex over the orbit category can be realized by a finite G-CW complex with isotropy in F. This is a generalization of Swan [32, Theorem A], which is based on a construction of Milnor [32, 3.1].

Associated to a G-CW complex X with isotropy in F, there is a chain complex of RΓ_{G}-modules defined by

C(X^{?}; R): ⋯ ⟶^{∂_{n+1}} R[X_{n} ^{?} ] ⟶^{∂_{n}} R[X_{n−1} ^{?} ] → ⋯ ⟶^{∂_{1}} R[X_{0} ^{?} ] → 0

where X_{i} denotes the set of i-dimensional cells in X and R[X_{i} ^{?} ] is the coefficient system with R[X_{i} ^{?} ](H) = R[X_{i}^{H}]. We denote the homology of this complex by H_{∗}(X^{?}; R), and in particular

H_{∗}(X^{?}; R)(H) = H_{∗}(X^{H}; R).

Given a chain complex C of RΓ_{G}-modules, there is a dimension function Dim C: F → Z, constant on conjugacy classes of subgroups, defined by

(Dim C)(H) = dim C(H),

for all H ∈ F, where the dimension of a chain complex of R-modules is defined in the usual way as the largest integer d such C_{d} ≠ 0.

It will be convenient to write (H) ≤ (K) whenever H^{g} ≤ K for some g ∈ G. Here (H) denotes the set of subgroups conjugate to H in G.

Definition 8.2. We call a function \underline{d}: F → Z monotone if it satisfies the property that \underline{d}(K) ⩽ \underline{d}(H) whenever (H) ≤ (K). We say that a monotone function \underline{d} is strictly monotone if \underline{d}(K) < \underline{d}(H), whenever (H) ≤ (K) and (H) ≠ (K). □

Note that \underline{d} monotone implies that \underline{d} is constant on conjugacy classes (such functions are usually called super class functions). We remark that the dimension function of a projective chain complex is always monotone: if (E_{H}P)(K) ≠ 0, then (E_{H}P)(L) ≠ 0 for every L ≤ K.

A chain complex C of RΓ_{G}-modules is connected if C is positive and H_{0}(C) = \underline{R}.

Definition 8.3. Let \underline{n}: F → Z be a monotone, non-negative function. A complex C of RΓ_{G}-modules is called an \underline{n}-Moore complex if it is connected, and for all H ∈ F, the reduced homology H̃_{i}(C(H)) = 0, for i ≠ \underline{n}(H). □

A special case of an \underline{n}-Moore complex is a homology \underline{n}-sphere.

Definition 8.4. We say that a complex C of RΓ_{G}-modules is an R-homology \underline{n}-sphere if it is an \underline{n}-Moore complex, and for all H ∈ F, we have H̃_{i}(C(H)) ≅ R, for i = \underline{n}(H). A homology \underline{n}-sphere is called oriented if the N_{G}(H)/H-action is trivial on the homology of C(H) for all H ∈ F.

The chain complex associated to the unit sphere X = S(V) of a real or complex representation V of G is an example of a Z-homology \underline{n}-sphere, where \underline{n}(H) = dim X^{H}. A G-CW complex X with this property is a homotopy representation in the sense of tom Dieck (see [10, Chap. II, Def. 10.1]), provided that its dimension function is strictly monotone. We will not use this terminology further.

We now introduce a technique to remove free modules above the homological dimension from a chain complex, without changing its chain homotopy type. For this delicate process we first need some algebraic lemmas.

Definition 8.5. Let Γ be an EI-category. A free RΓ -module F is called isotypic of type x ∈ Ob(Γ) if it is isomorphic to a direct sum of copies of the free module E_{x}R[x].

For extensions involving isotypic modules we have a splitting property.

Lemma 8.6. Let

E : 0 → F → F′ → M → 0

be a short exact sequence of RΓ -modules over an EI-category Γ, such that both F and F′ are isotypic free modules of the same type x ∈ Ob(Γ). If M(x) is R-torsion free, then E splits and M is stably free.

Proof. It is enough to prove the result in case F = E_{x}R[x], where x ∈ Ob Γ. The general case follows from this by an easy induction. Consider the extension

E : 0 → E_{x}R[x] ⟶^{j} F → M → 0 .

By the adjointness property

Hom_{RΓ} (E_{x}R[x], N) ≅ Hom_{R[x]}(R[x], N(x))

for any RΓ -module N. We apply this to the given injection j: E_{x}R[x] → F′ = (E_{x}R[x])^{m}. Since

E(x): 0 → R[x] ⟶^{j} R[x]^{m} → M(x) → 0

has R-torsion free cokernel M(x), this sequence splits over R[x]. By the naturality of the adjointness property, we get a splitting of j over RΓ. □

Recall that hdim C(H) denotes the homological dimension of the chain complex C(H).

Proposition 8.7. Let C be a finite free chain complex of RΓ_{G}-modules, and let H ∈ F have the property that hdim C(H) < d := dim C(H). Suppose that dim C(K) ⩽ (d − 2) for all (H) ≤ (K), (H) ≠ (K). Then C ≃ D, where D is a finite free complex with dim D(H) = d − 1, and dim D(K) = dim C(K) for all (K) ≠ (H).

Proof. Consider the subcomplex C′ of C formed by free summands of C isomorphic to Z[G/K^{?} ], with (G/K)^{H} ≠ 0 or equivalently (H) ≤ (K). The boundary maps of C′ are the restrictions of the usual boundary maps to these submodules. Since dim C(K) ⩽ (d − 2) for all (H) ≤ (K) such that (H) ≠ (K), the free modules C′_{d} and C′_{d−1} are isotypic of type G/H. We have

C′ : 0 → C′_{d} → C′_{d−1} → ⋯ → C′_{1} → C′_{0} → 0

where d = dim C(H). Note that C(H) = C′(H), so the map ∂_{d}: C′_{d} → C′_{d−1} is injective by the condition that hdim C(H) < dim C(H). Now we can apply Lemma 8.6 to the extension

0 → C′_{d} ⟶^{∂_{d}} C′_{d−1} → coker ∂_{d} → 0

and conclude that coker(∂_{d}) is a stably free RΓ_{G}-module. By adding elementary chain complexes to C of the form Z[G/H^{?} ] ⟶^{id} Z[G/H^{?} ] in the adjacent dimensions (d − 1) and (d − 2), we can assume that coker(∂_{d}) is free. Consider the diagram

[figure: ⋯; 0; C′_{d}; id; id; C′_{d}; ∂_{d}; 0; ⋯; 0; 0; C′: ⋯; 0; C′_{d}; C′_{d−1}; C′_{d−2}; ⋯; C′_{0}; 0; D′: ⋯; 0; 0; coker ∂_{d}; C′_{d−2}; …; C′_{0}; 0]

The chain complex D′ is a chain complex of free modules and it is chain homotopy equivalent to C′. Now define D as the push-out in the the following diagram:

[figure: ker; ker; C′; C; C/C′; D′; D; C/C′]

Since, C′ and D′ are chain homotopy equivalent, then C and D are chain homotopy equivalent. Also, note that dim D(H) = dim D′(H) = (d−1), and dim D(K) = dim C(K) for all (K) ≠ (H). □

This immediately gives the following.

Corollary 8.8. Let C be a finite free chain complex of RΓ_{G}-modules. Suppose that C is a homology \underline{n}-sphere, with \underline{n} strictly monotone. Then C is chain homotopy equivalent to a complex D with Dim D = \underline{n}.

Proof. Since C is a homology \underline{n}-sphere, \underline{n}(K) = hdim C(K), for all K ∈ F. We apply the previous result to a subgroup H, which is maximal with respect to the property that hdim C(H) < d := dim C(H). Then \underline{n}(K) = dim C(K) for all K ∈ F larger than H. Since \underline{n} is strictly monotone, dim C(K) ⩽ (d − 2) for all (H) ≤ (K), (H) ≠ (K). This process can be repeated until Dim D = \underline{n}. □

When the dimension function of C is not strictly monotone, we get a weaker result. Following Section 2, we define l(H, K) as the maximum length of a chain of conjugacy classes of subgroups

(H) = (H_{0}) ≨ (H_{1}) ≨ . . . . . . ≨ (H_{l}) = (K)

where all H_{i} ∈ F, 0 ⩽ i ⩽ l.

Corollary 8.9. Let C be a finite free chain complex of RΓ_{G}-modules, and let \underline{n}: F → Z be a monotone function such that hdim C(H) ⩽ \underline{n}(H) for all H ∈ F. Assume that l(H, K) ⩽ k whenever \underline{n}(H) = \underline{n}(K). Then, C is chain homotopy equivalent to a complex D which satisfies D_{i}(H) = 0 for all i > \underline{n}(H) + k.

Proof. Let

(H) = (H_{0}) ≨ (H_{1}) ≨ . . . . . . ≨ (H_{l}) = (K)

be a maximal length chain of subgroups in F with \underline{n}(H) = \underline{n}(K). Since \underline{n} is monotone, \underline{n}(H_{i}) = \underline{n}(H) for 0 ⩽ i ⩽ l. By repeated application of Proposition 8.7, working down from the maximal element K, we can obtain dim C(H_{l−i}) = \underline{n}(H) + i, for 0 ⩽ i ⩽ l. Since l = l(H, K) ⩽ k, we have dim C(H) ⩽ \underline{n}(H) + k as required. □

The main purpose of this section is to prove the following theorem:

Theorem 8.10 (Pamuk [27]). Let C be a finite free chain complex of ZΓ_{G}-modules. Suppose C is an \underline{n}-Moore complex such that \underline{n}(H) ⩾ 3 for all H ∈ F. Suppose further that C_{i}(H) = 0 for all i > \underline{n}(H) + 1, and all H ∈ F. Then there is a finite G-CW complex X, such that C(X^{?}; Z) is chain homotopy equivalent to C, as chain complexes of ZΓ_{G}-modules.

Note that the resulting complex X will have isotropy in F. We first prove a lemma (compare [20, Thm. 13.19]).

Lemma 8.11. Let X be a finite G-CW complex. Suppose that we are given a free ZΓ_{G}- module F, and a ZΓ_{G}-module homomorphism φ: F → H_{n}(X^{?}; Z), for some n ⩾ 2. Assume further that X^{H} is (n − 1)-connected for every H ∈ F such that Z[G/H^{?} ] is a summand of F. Then, by attaching (n + 1)-cells to X, we can obtain a G-CW complex Y such that

H_{i}(X^{?}; Z) ≅ H_{i}(Y^{?}; Z) for i ≠ n, n + 1,

and

0 → H_{n+1}(X^{?}; Z) → H_{n+1}(Y^{?}; Z) → F ⟶^{φ} H_{n}(X^{?}; Z) → H_{n}(Y^{?}; Z) → 0

is exact.

Proof. Let Z be a wedge of n-spheres with a G action on them such that H̃_{n}(Z^{?}; Z) ≅ F as ZΓ_{G}-modules. We want to construct a map f: Z → X realizing φ. But H_{n}(X^{H}; Z) ≅ π_{n}(X^{H}), for every H ∈ F such that Z[G/H^{?} ] is a summand of F, since X^{H} is assumed to be (n − 1)-connected. Therefore, we can represent the images of an Z[N_{G}(H)/H]-basis under φ for the isotypic summand in F of type G/H by maps f_{i}: S^{n} → X^{H}. We extend these maps equivariantly to maps f̄_{i}: S^{n} × G/H → X. By repeating this construction for each type G/H in F, we obtain an equivariant map f: Z → X realizing φ. Take Y to be the mapping cone of f. Then, it is easy to see that Y satisfies the desired conditions. □

We also need the following lemma:

Lemma 8.12. Let C be a finite free chain complex of ZΓ_{G}-modules. Suppose that C is connected, and H_{i}(C) = 0, for i = 1, 2. Then, C is chain homotopy equivalent to a complex of the form

⋯ → C_{n} → C_{n−1} → ⋯ → C_{3} → C_{2}(X) → C_{1}(X) → C_{0}(X) → 0

where C_{2}(X) → C_{1}(X) → C_{0}(X) → 0 is the initial part of the chain complex C(X^{?}; Z), for some G-CW complex X with isotropy in F, and X^{H} simply-connected for all H ∈ F.

Proof. There is a G-CW complex E_{F}G satisfying the properties

(i) All isotropy subgroups of E_{F}G are in F, (ii) For every H ∈ F, the fixed point set (E_{F}G)^{H} is contractible [22, Theorem 1.9].

The chain complex D := C((E_{F}G)^{?}; Z) of this space gives a free resolution of \underline{Z} as a ZΓ_{G}-module. Since H_{i}(C) = 0, for i = 1, 2, the following sequences are both exact

(8.13)  [figure: 0; A; C_{2}; ∂_{2}^{C}; C_{1}; ∂_{1}^{C}; C_{0}; \underline{Z}; 0; 0; B; D_{2}; ∂_{2}^{D}; D_{1}; ∂_{1}^{D}; D_{0}; \underline{Z}; 0]

where A = ker ∂_{2}^{C} and B = ker ∂_{2}^{D}.

By an elementary operation on a sequence A → C_{2} → C_{1} → C_{0} we mean adding or removing trivial free summands F ⟶^{id} F in adjacent dimensions. It is clear that elementary operations don’t change the chain homotopy type of the upper and lower sequences in diagram (8.13).

Then, by Schanuel’s Lemma [32, 1.1], there exist free modules F and F′ such that A ⊕ F ≅ B ⊕ F′. In fact, the argument in Schanuel’s lemma can be extended to say that the induced isomorphism γ: A ⊕ F ≅ B ⊕ F′ comes from a chain isomorphism after a sequence of elementary operations (compare [20, p. 279]).

In other words, there exists a chain isomorphism

(8.14)  [figure: 0; A ⊕ F; ≅; γ; C_{2} ⊕ F_{2}; f_{2}; ≅; C_{1} ⊕ F_{1}; f_{1}; ≅; C_{0} ⊕ F_{0}; f_{0}; ≅; \underline{Z}; 0; 0; B ⊕ F′; D_{2} ⊕ F′_{2}; D_{1} ⊕ F′_{1}; D_{0} ⊕ F′_{0}; \underline{Z}; 0]

for some suitable choices of free modules, where the upper and lower sequences in diagram (8.14) are obtained from those in diagram (8.13) by elementary operations (see Proposition 3.3.3 in [27]).

In the first step, we stabilize (A → C_{2}) ↦ (A ⊕ F → C_{2} ⊕ F), by adding the identity on F, and similarly (B → D_{2}) ↦ (B ⊕ F′ → D_{2} ⊕ F′). We therefore have a chain map

[figure: 0; A ⊕ F; ≅; γ; C_{2} ⊕ F; C_{1}; C_{0}; \underline{Z}; 0; 0; B ⊕ F′; D_{2} ⊕ F′; D_{1}; D_{0}; \underline{Z}; 0]

which is a chain homotopy equivalence (by composition with the chain map in (8.14)). After an elementary operation on C, we can use the isomorphism γ: A ⊕ F ≅ B ⊕ F′ to splice the bottom sequence to C, and obtain a chain homotopy equivalence

[figure: ⋯; C_{4}; C_{3} ⊕ F; C_{2} ⊕ F; C_{1}; C_{0}; \underline{Z}; 0; ⋯; C_{4}; C_{3} ⊕ F; D_{2} ⊕ F′; D_{1}; D_{0}; \underline{Z}; 0]

The top sequence is chain homotopy equivalent to C, so to complete the proof we need to show that the sequence D_{2} ⊕ F′ → D_{1} → D_{0} → 0 can be realized as the first three terms of a chain complex of a G-CW complex X, with isotropy in F, such that X^{H} is simply connected for all H ∈ F: since E_{F}G is contractible, using Lemma 8.11, we can attach free 2-cells to its two skeleton E_{F}G^{(2)}. The resulting complex X will have the desired properties. □

Now, we are ready to prove Theorem 8.10.

Proof of Theorem 8.10. We can assume that the complex C is of the form given in Lemma 8.12. We obtain a map φ: C_{3} → C_{2}(X^{(2)}) which induces an isomorphism Z_{2}(C) → Z_{2}(X^{(2)}) between 2-cycles of these chain complexes. This is the starting point for an inductive argument based on applying Lemma 8.11 at each step.

Fix n ⩾ 2, and assume by induction that there is an n-dimensional G-CW complex X^{(n)}, and a chain map

\xymatrix{⋯ \ar[r] & C_{n+2} \ar[r] \ar[d] & C_{n+1} \ar[r] \ar[d] & C_{n} \ar[r] \ar[d] & … \ar[r] & C_{1} \ar[r] \ar[d] & C_{0} \ar[r] \ar[d] & 0 \\ ⋯ \ar[r] & 0 \ar[r] & Z_{n}(X^{(n)}) \ar[r] & C_{n}(X^{(n)}) \ar[r] & … \ar[r] & C_{1}(X^{(n)}) \ar[r] & C_{0}(X^{(n)}) \ar[r] & 0}

which induces an homology isomorphism for dimensions less than or equal to (n − 1), and at dimension n the induced map Z_{n}(C) → Z_{n}(X^{(n)}) is an isomorphism.

Note that dim C(H) ⩽ \underline{n}(H) + 1 by assumption. If Z[G/H^{?} ] is a summand of C_{n+1}, then (n+1) ⩽ dim C(H) ⩽ \underline{n}(H)+1 implies \underline{n}(H) ⩾ n, and hence the H-fixed set of X^{(n)} is (n − 1)-connected. We can now apply Lemma 8.11 to the map φ: C_{n+1} → H_{n}(X^{(n)}; Z) defined by the composition

ϕ: C_{n+1} → Z_{n}(C) ≅ Z_{n}(X^{(n)}) → H_{n}(X^{(n)}; Z).

Let us call the resulting complex X^{(n+1)}. Note that there is a chain map C → C(X^{(n+1)}) which induces an isomorphism on homology for dimensions ⩽ n, and at dimension n + 1 we have an isomorphism Z_{n+1}(C) → Z_{n+1}(X^{(n+1)}). Since C is finite dimensional, after finitely many steps, we will obtain a finite dimensional G-CW complex X and a chain map f: C → C(X) which induces isomorphism on homology for all dimensions. Since both C and C(X) are free ZΓ_{G}-chain complexes, f is a chain homotopy equivalence as desired. □

9. The proof of Theorem A

Let G = S_{5}, the symmetric group of order 120 permuting {1, 2, 3, 4, 5}, and let S_{4} ≤ G denote the permutations fixing {5}. We work relative to the family F of rank 1 subgroups of 2-power order. Let Γ_{G} = Or_{F} G. Our family F consists of the subgroups of G which are conjugate to one the subgroups in the set

{1, C_{2}^{A}, C_{2}^{B}, C_{4}}

where C_{2}^{A} = ⟨(12)(34)⟩, C_{2}^{B} = ⟨(12)⟩, and C_{4} = ⟨(1234)⟩. In addition we will consider the Sylow subgroups C_{3} = ⟨(123)⟩ and C_{5} = ⟨(12345)⟩. It is convenient to note that for H = S_{4} ≤ G, we have

N_{H}(C_{4}) = D_{8} = N_{H}(C_{2}^{A}) = N_{G}(C_{4}),

while N_{H}(C_{2}^{B}) = E = ⟨(12), (34)⟩, and N_{H}(C_{3}) = S_{{123}}. On the other hand, N_{G}(C_{2}^{B}) = ⟨(12), S_{{345}}⟩ and N_{G}(C_{3}) = S_{{123}} × ⟨(45)⟩.

Our strategy will be to construct finite projective complexes C^{(p)} with isotropy in F over Z_{(p)}Γ_{G}, for each prime p dividing the order of |G|, which are R-homology \underline{n}-spheres with respect to the the same homology dimension function \underline{n}. The gluing theory of Section 6, Theorem 6.7, will be used to construct a finite projective Z-homology \underline{n}-sphere over ZΓ_{G} from this data. Then the join construction from Section 7 will allow us to find a finite free complex, to which the realization theorem of Section 8 will apply.

We introduce the notation \underline{R}_{0} for the RΓ_{G}-module defined by \underline{R}_{0}(K) = 0, for K ≠ 1, and \underline{R}_{0}(1) = R with trivial G-action. In other words, \underline{R}_{0} = I_{1}(R) as defined in Section 2.

9A. The case p = 2. Let H = S_{4} ≤ G, R = Z_{(2)} and consider the standard H-action on the 2-sphere given by the rotational symmetries of the octahedron. Let X denote the H-CW complex associated to the first barycentric subdivision of the octahedron. Then X has isotropy in the family consisting of the cyclic subgroups of H of orders ≤ 4.

Let Γ_{H} = Or_{F} H denote the orbit category for H with respect to the family F_{H} = F∩H. Consider the chain complex C(X^{?}; R) as a chain complex of RΓ_{H}-modules, by restricting this functor to the full subcategory Γ_{H} of the orbit category Or(H). This gives an exact sequence of the form

0 → \underline{R}_{0} → 2R[H/1^{?} ] → 3R[H/1^{?} ] → R[H/C_{4} ^{?} ] ⊕ R[H/C_{2}^{B} ^{?} ] ⊕ R[H/C_{3} ^{?} ] → H_{0} → 0,

where all the modules in the extension (excluding the ends) except R[H/C_{3} ^{?} ] are free RΓ_{H}-modules, and H_{0} = H_{0}(C(X^{?}; R)).

Since R[H/C_{3}] is a projective RH-module (it is induced up from R, which is projective over R[C_{3}]), we see that R[H/C_{3} ^{?} ] = I_{1}R[H/C_{3}] as RΓ_{H}-modules. Therefore C(X^{?}; R) is a finite projective chain complex over RΓ_{H}.

It is useful to represent an RΓ_{H}-module M by a labelled tree diagram:

M = [figure: M(C_{4}); M(C_{2}^{A}); M(C_{2}^{B}); M(1)]

with one vertex for each isomorphism class of objects, and edges given by the partial ordering of the subgroups in F up to conjugacy. The labels are given by the R[N_{H}(K)/K]- modules M(K), for K ∈ F.

For the homology module H_{0}(X^{?}; R) over the orbit category Γ_{H} we have the diagram

H_{0} = [figure: R[D_{8}/C_{4}]; R[D_{8}/C_{4}]; R[E/C_{2}^{B}]; R]

The (k+1)-fold join of C(X^{?}; R) with itself (see Section 7) is a finite projective complex of the form

C : 0 → \underline{R}_{0} → C_{n} → ⋯ → C_{k} → ⋯ → C_{0} → \underline{R} → 0

over RΓ_{H} with (n + 1) = 3(k + 1). In case X = S(V), where V ≅ R^{3} is a real orthogonal H-representation, then the join construction on spheres just produces the unit sphere S(V ⊕ ⋯ ⊕ V) in the direct sum of (k + 1) copies of V. This sphere has real dimension n = 3(k + 1) − 1. The purpose of the join construction is to produce a complex with dimension gaps between the non-zero homology groups, as required by Theorem 6.7 for glueing the different primes together.

We have H_{0}(C) = \underline{R} and H_{n} = \underline{R}_{0}. If (k + 1) is even, then H_{k}(C(Q)) = R, with trivial N_{H}(Q)/Q-action, and H̃_{i}(C(Q)) = 0, for i ≠ k, for each non-trivial Q ∈ F. By Proposition 6.4, we obtain a chain complex C^{(2)} of projective RΓ_{G}-modules, having homology isomorphic to R, with trivial N_{G}(Q)/Q-action. By construction, the homology dimension function \underline{n} for C^{(2)} is the same as for C. Notice that \underline{n} is monotone, but not strictly monotone.

9B. The case p = 3. Let R = Z_{(3)} and K = C_{2}^{B}. The 3-period of G = S_{5} is four [7, Chap. XII, Ex. 11], so by Swan [32] there exists a periodic projective resolution P with

0 → R → P_{n} → ⋯ → P_{1} → P_{0} → R → 0

over the group ring RG, for any n such that 4 | (n + 1). We will assume that 12 | (n + 1), and let k be defined by the equation (n + 1) = 3(k + 1). Similarly, since N_{G}(K)/K ≅ S_{3} also has 3-period 4, we have a chain complex D yielding a periodic projective resolution

0 → R → D_{k} → ⋯ → D_{0} → R → 0

over RS_{3}. In the rest of this section we let W_{K} = N_{G}(K)/K to simplify the notation.

We want a finite projective chain complex C over RΓ_{G} which fits into an extension of chain complexes

0 → E_{1}P → C → I_{K}D → 0

where the induced exact sequence on the 0-th homology

0 → \underline{R}_{0} → H_{0}(C) → I_{K}R → 0

is the non-trivial extension ot tree diagrams (with vertices at {1, K})

0 → \begin{matrix}0 \\ R \end{matrix} → R_{∨}_{id} R → \begin{matrix}R \\ 0 \end{matrix} → 0.

For a projective R[W_{K}]-module D, the module I_{K}D has a finite projective resolution of the form

\xymatrix{0 \ar[r] & E_{1} Res_{1} E_{K}D \ar[r] & E_{K}D \ar[r] & I_{K}D \ar[r] & 0} .

By definition of the functors E_{x} and I_{x} (see Section 2), the canonical map

E_{x}M → I_{x}M → 0

is always surjective for any R[x]-module M. We have E_{K}R[W_{K}] = R[G/K^{?} ] and hence E_{K}D is projective. Also Res_{1} E_{K}D is projective, because it is a summand of R[G/K] which is projective as an Z_{(3)}G-module. This shows that, once constructed, C will be homotopy equivalent to a finite projective chain complex by Lemma 6.2.

Associated to every RG-chain map f: Res_{1} E_{K}D → P, there is a chain complex C which fits into the push-out diagram

[figure: 0; E_{1} Res_{1} E_{K}D; E_{1}f; E_{K}D; I_{K}D; 0; 0; E_{1}P; C; I_{K}D; 0 .]

We want to choose f so that C satisfies the condition on homology. Note that

H_{0}(Res_{1} E_{K}D) = Res_{1} E_{K}R = R[G/N_{G}(K)].

Since the modules Res_{1} E_{K}D_{i} = (E_{K}D_{i})(1) are projective for all i and P is exact, there exists a chain map f: Res_{1} E_{K}D → P

\xymatrix{⋯ \ar[r] & (E_{K}D_{1})(1) \ar[d]^{f_{1}} \ar[r] & (E_{K}D_{0})(1) \ar[d]^{f_{0}} \ar[r] & R[G/N_{G}(K)] \ar[d]^{ε} \ar[r] & 0 \\ ⋯ \ar[r] & P_{1} \ar[r] & P_{0} \ar[r] & R \ar[r] & 0}

lifting the augmentation map R[G/N_{G}(K)] ⟶^{ε} R. To see that the resulting push-out complex C has the desired properties, consider the homology at zero for the diagram of chain complexes given above. Since I_{K} is an exact functor, H_{1}(I_{K}D) = I_{K}H_{1}(D) = 0, and we get

[figure: 0; H_{0}(E_{1} Res_{1} E_{K}D); H_{0}(E_{1}f); H_{0}(E_{K}D); H_{0}(I_{K}D); 0; 0; H_{0}(E_{1}P); H_{0}(C); H_{0}(I_{K}D); 0]

where H_{0}(E_{1}P) = E_{1}R. Note that

H_{0}(E_{1} Res_{1} E_{K}D) = E_{1} Res_{1} H_{0}(E_{K}D) = E_{1} Res_{1} E_{K}R = E_{1}R[G/N_{G}(K)].

This gives a diagram of the form

[figure: E_{1} ker ε; E_{1} ker ε; 0; E_{1}R[G/N_{G}(K)]; E_{1}(ε); E_{K}R; I_{K}R; 0; 0; E_{1}R; H_{0}(C); I_{K}R; 0]

where the middle vertical sequence of RΓ_{G}-modules is given by

0 → \begin{matrix}0 \\ ker ε \end{matrix} → \begin{matrix}R \\ R[G/N_{G}(K)] \end{matrix} → R_{∨}_{id} R → 0

This shows that H_{0}(C) has the desired form.

Now, to obtain the same homology dimension function as for the complex C^{(2)}, more homology must be added to the complex C. We need to extend H_{k} and H_{0} via the non-split extensions

0 → H_{0} → Ĥ_{0} → N → 0 and 0 → H_{k} → Ĥ_{k} → N → 0

where

N = [figure: R; id; R; 0; 0]

The module N has a finite projective resolution of the form

0 → E_{1}R[G/D_{8}] → E_{C_{4}}R → N → 0.

Note that Res_{1} E_{C_{4}}R = R[G/N_{G}(C_{4})] = R[G/D_{8}], and for Q = C_{2}^{A} we have

Res_{Q} E_{C_{4}}R = R ⊗_{R[D_{8}/C_{4}]} R[(G/C_{4})^{Q}] = R ⊗_{R[D_{8}/C_{4}]} R[N_{G}(Q)/N_{C_{4}}(Q)] = R

where the equality in the middle comes from Lemma 3.6. Since R is projective as an R[D_{8}/C_{4}]-module, E_{C_{4}}R is projective. It is easy to see that E_{1}R[G/D_{8}] is also projective. So, by Proposition 6.8, we can replace C with a finite projective chain complex C^{(3)} over RΓ_{G} which has the desired homology.

9C. The case p = 5. For p = 5, the situation is easier than the case p = 3. Let R = Z_{(5)}. The 5-period of S_{5} equals 8, so by Swan [32] there exists a periodic projective resolution P over the group ring RG, giving an exact sequence

0 → R → P_{n} → ⋯ → P_{1} → P_{0} → R → 0

for any positive integer n such that n + 1 = 3(k + 1) for some integer k, with 8 | (k + 1). We start with the RΓ_{G}-complex C = E_{1}P obtained by the extension functor from P. Since C has no homology at the non-trivial 2-subgroups in F, we need to change the homology at H_{0} and at H_{k} to match the homology we have for p = 2 and p = 3. Note that we need to extend H_{k} and H_{0} via the non-split extensions

0 → H_{0} → Ĥ_{0} → M → 0 and 0 → H_{k} → Ĥ_{k} → M → 0

where

M = [figure: R; id; R; R; 0]

Let K = C_{2}^{B}. The module M is the direct sum of L (which has the same form as N) and I_{K}R. We claim that each of these modules have finite projective resolutions over RΓ_{G}. For I_{K}R we have a resolution of the form

0 → E_{1}R[G/(K × S_{3})] → E_{K}R → I_{K}R → 0.

Note that

Res_{1} E_{K}R = R[G/N_{G}(K)] = R[G/(K × S_{3})]

where S_{3} denotes the subgroup of S_{5} generated by symmetries of {3, 4, 5}. Since R is projective as an R[N_{G}(K)/K]-module, E_{K}R is projective. It is clear that E_{1}R[G/(K×S_{3})] is also projective. So, the above resolution is a projective resolution of I_{K}R. We can also write a finite projective resolution for L (similar to the resolution given for N). So, by Proposition 6.8, we can replace C with a finite projective chain complex C^{(5)} which has the desired homology.

The proof of Theorem A. We will first construct a projective chain complex C over ZΓ_{G} with isotropy in F, by applying Theorem 6.7 to glue the p-local complexes C^{(p)}, for p = 2, 3, 5. Note that in the constructions of C^{(p)} above, we may choose any integer k such that k odd, n+1 = 3(k +1), 12 | (n+1) and 8 | (k +1). To satisfy the first condition in Theorem 6.7, that the distance between non-zero homology groups of the C^{(p)} is larger than l(Γ_{G}) = 2, we will also need k ≥ 3 and n − k ≥ 3.

Remark 9.1. The minimum value for k satisfying the requirements used above is k = 7, which gives n = dim C = 23.

The ZΓ_{G}-module H needed to satisfy the second condition in Theorem 6.7 is given by H_{i}(K) = Z, for i = 0, \underline{n}(K) with K ∈ F, and zero otherwise. By Lemma 6.2, C is chain homotopy equivalent to a finite projective complex. To obtain a finite free complex, we can apply Theorem 7.6, which (possibly after some joins) produces a finite free ZΓ_{G}-chain complex C with the Z-homology of an \underline{n}-sphere, and \underline{n}(K) ⩾ 3 for all K ∈ F.

Note that our homology dimension function \underline{n} is not strictly monotone, since \underline{n}(C_{2}^{A}) = \underline{n}(C_{4}), but by Corollary 8.9 we can modify our complex to satisfy the conditions for geometric realization in Theorem 8.10, since l(C_{2}^{A}, C_{4}) = 1. Applying Theorem 8.10, we conclude that G = S_{5} acts on a finite G-CW complex X with isotropy in F. □

Remark 9.2. For this particular example we needed to apply Theorem 7.6 with one join tensor operation, because K̃_{0}(ZΓ_{G}) = Z/2. This follows from Theorem 7.8, Lemma 7.5 and well-known calculations showing that K̃_{0}(Z[N_{G}(Q)/Q]) = 0, for 1 ≠ Q ∈ F, but K̃_{0}(Z[G]) = Z/2. Note that, by Dress induction, it is enough to consider the projective class groups of p-hyperelementary subgroups of G (see [9, §50], [28]). We therefore obtain a finite G-CW complex X ≃ S^{47} with isotropy in F.

References

[1] A. Adem and J. H. Smith, Periodic complexes and group actions, Ann. of Math. (2) 154 (2001), 407–435.

[2] D. J. Benson, Representations and cohomology. I, second ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998, Basic representation theory of finite groups and associative algebras.

[3] , Representations and cohomology. II, second ed., Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1998, Cohomology of groups and modules.

[4] 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.

[5] 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.

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

[7] H. Cartan and S. Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999, With an appendix by David A. Buchsbaum, Reprint of the 1956 original.

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

[9] 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.

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

[11] A. Dold, Zur Homotopietheorie der Kettenkomplexe, Math. Ann. 140 (1960), 278–298.

[12] R. M. Dotzel and G. C. Hamrick, p-group actions on homology spheres, Invent. Math. 62 (1981), 437–442.

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

[14] J. Grodal, Higher limits via subgroup complexes, Ann. of Math. (2) 155 (2002), 405–457.

[15] J. Grodal and J. H. Smith, Algebraic models for finite G–spaces, Oberwolfach Reports, vol. 4, Eur. Math. Soc., Zürich, 2007, Abstracts from the Homotopy Theory workshop, September 16–22, 2007, organized by Paul Goerss, John Greenlees and Stefan Schwede, pp. 2690–2692.

[16] I. Hambleton, L. R. Taylor, and E. B. Williams, Dress Induction and the Burnside quotient Green ring, Algebra & Number Theory 3 (2009), 511–541.

[17] 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.

[18] M. A. Jackson, Qd(p)-free rank two finite groups act freely on a homotopy product of two spheres, J. Pure Appl. Algebra 208 (2007), 821–831.

[19] T.-Y. Lam, Induction theorems for Grothendieck groups and Whitehead groups of finite groups, Ann. Sci. École Norm. Sup. (4) 1 (1968), 91–148.

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

[21] , Chern characters for proper equivariant homology theories and applications to K- and Ltheory, J. Reine Angew. Math. 543 (2002), 193–234.

[22] , Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 269–322.

[23] S. Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1975 edition.

[24] , Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998.

[25] G. Mislin, On group homomorphisms inducing mod-p cohomology isomorphisms, Comment. Math. Helv. 65 (1990), 454–461.

[26] B. Oliver and Y. Segev, Fixed point free actions on Z-acyclic 2-complexes, Acta Math. 189 (2002), 203–285.

[27] S. Pamuk, Periodic resolutions and finite group actions, Ph.D. thesis, McMaster University, 2008.

[28] I. Reiner, Projective class groups of symmetric and alternating groups, Linear and Multilinear Algebra 3 (1975/76), 147–153, Collection of articles dedicated to Olga Taussky-Todd.

[29] D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700–712.

[30] E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York, 1966.

[31] R. G. Swan, Induced representations and projective modules, Ann. of Math. (2) 71 (1960), 552–578.

[32] , Periodic resolutions for finite groups, Ann. of Math. (2) 72 (1960), 267–291.

[33] P. Symonds, Mackey functors and control of fusion, Bull. London Math. Soc. 36 (2004), 623–632.

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

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

Department of Mathematics, McMaster University, Canada

  E-mail address: hambleton@mcmaster.ca

Department of Mathematics, McMaster University, Canada

  E-mail address: pamuks@math.mcmaster.ca

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

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