The Segal Conjecture for Infinite Groups

TL;DR

This paper extends the Segal Conjecture to infinite groups, establishing an isomorphism between stable cohomotopy groups under specific geometric conditions, thus broadening its applicability beyond finite groups.

Contribution

It formulates and proves a version of the Segal Conjecture for infinite groups with finite models for their classifying spaces, generalizing the original conjecture.

Findings

For hyperbolic groups, the zero-th stable cohomotopy of BG matches the I-adic completion of the equivariant stable cohomotopy of underline{E}G.

The conjecture holds for groups with finite models for their classifying space for proper actions.

The result connects stable cohomotopy of classifying spaces with algebraic completions in a broader group context.

Abstract

We formulate and prove a version of the Segal Conjecture for infinite groups. For finite groups it reduces to the original version. The condition that G is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space underline{E}G for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups G that the zero-th stable cohomotopy of the classifying space BG is isomorphic to the I-adic completion of the ring given by the zero-th equivariant stable cohomotopy of underline{E}G for I the augmentation ideal.

Equations

$${\pi }_G^m(X)\hat{{}_{{\mathbb{I}}_G}}\stackrel{\cong }{\to }{\pi }_s^m(EG{\times }_{G}X).$$

$$A(G)\hat{{}_{{\mathbb{I}}_G}}\stackrel{\cong }{\to }{\pi }_s^0(BG).$$

$${\lambda }_G^m(X):{\left\{{\pi }_G^m(X)/{\mathbb{I}}_G(L{)}^n\cdot {\pi }_G^m(X)\right\}}_{n\ge 1}\stackrel{\cong }{\to }{\left\{{\pi }_s^m\left((EG{\times }_{G}X{)}_{(n-1)}\right)\right\}}_{n\ge 1}$$

$${\pi }_G^m(X)\hat{{}_{{\mathbb{I}}_G(L)}}\cong {\pi }_s^m(EG{\times }_{G}X).$$

$${\pi }_G^m(\underline{E}G)\hat{{}_{{\mathbb{I}}_G(\underline{E}G)}}\cong {\pi }_s^m(BG).$$

$$A(G)\widehat{{}_{I_G}}\cong {\pi }_s^0(BG).$$

$$\cup :{\mathcal{H}}_G^m(X,A){\otimes }_R{\mathcal{H}}_G^n(X,B)$$

$${\mathrm{ind}}_{\alpha }:{\mathcal{H}}_G^n({\mathrm{ind}}_{\alpha }(X,A))$$

$${\mathcal{H}}_G^n(G/K)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K^{n}_{G}(G/H)=K^{n}_{H}(\ast)=\left\{\begin{array}[]{lll}R_{{\mathbb{C}}}(H)&&n\text{ even};\\ \{0\}&&n\text{ odd},\end{array}\right.$$

$${\pi }_n^H={\pi }_{-n}^H=\underset{(K)}{⨁}{\pi }_{-n}^s(B{W}_{H}K),$$

$$\begin{array}[]{lllll}|\pi^{n}_{G}(G/H)|&<&\infty&&n\leq-1;\\ \pi^{0}_{G}(G/H)&=&A(H);&&\\ \pi^{n}_{G}(G/H)&=&\{0\}&&n\geq 1,\end{array}$$

$$M_0\stackrel{{\alpha }_1}{\leftarrow }{M}_1\stackrel{{\alpha }_2}{\leftarrow }{M}_2\stackrel{{\alpha }_3}{\leftarrow }{M}_3\stackrel{{\alpha }_4}{\leftarrow }\dots .$$

$$\{M_n,{\alpha }_n\}\stackrel{\{f_n\}}{\to }\{M_n^{\prime },{\alpha }_n^{\prime }\}\stackrel{g_n}{\to }\{M_n^{\prime \prime },{\alpha }_n^{\prime \prime }\}$$

$$\begin{gathered} 0\to {\mathrm{invlim}}_{n\ge 1}{M}_n^{\prime }\stackrel{{\mathrm{invlim}}_{n\ge 1}{f}_n}{\to }{\mathrm{invlim}}_{n\ge 1}{M}_n\stackrel{{\mathrm{invlim}}_{n\ge 1}{g}_n}{\to }{\mathrm{invlim}}_{n\ge 1}{M}_n^{\prime \prime }\stackrel{\delta }{\to } \\ {\mathrm{invlim}}_{n\ge 1}^1{M}_n^{\prime }\stackrel{{\mathrm{invlim}}_{n\ge 1}^1{f}_n}{\to }{\mathrm{invlim}}_{n\ge 1}^1{M}_n\stackrel{{\mathrm{invlim}}_{n\ge 1}^1{g}_n}{\to }{\mathrm{invlim}}_{n\ge 1}^1{M}_n^{\prime \prime }\to 0. \end{gathered}$$

$$\begin{array}[]{llcl}{\operatorname{invlim}}_{n\geq 1}f_{n}\colon&{\operatorname{invlim}}_{n\geq 1}M_{n}&\xrightarrow{\cong}&{\operatorname{invlim}}_{n\geq 1}N_{n};\\ {\operatorname{invlim}}_{n\geq 1}^{1}f_{n}\colon&{\operatorname{invlim}}_{n\geq 1}^{1}M_{n}&\xrightarrow{\cong}&{\operatorname{invlim}}_{n\geq 1}^{1}N_{n}.\end{array}$$

$$\{M^{\prime }/I^n{M}^{\prime }\}\to \{M/I^{n}M\}\to \{M^{\prime \prime }/I^n{M}^{\prime \prime }\}$$

$${\mathcal{H}}_G^n(X)\stackrel{{\prod }_{\overline{x}\in G\backslash {\pi }_0(X)}{\mathrm{ind}}_{\{1\}\to G}\circ {\mathcal{H}}_G^n(x)}{\to }\prod _{\overline{x}\in G\backslash {\pi }_0(X)}{\mathcal{H}}^n(\ast ).$$

$$\begin{gathered} {\mathbb{I}}_G(L{)}^n\cdot {\mathcal{H}}_G^m(X)\stackrel{i}{\to }{\mathcal{H}}_G^m(X)\stackrel{{\mathcal{H}}_G^m(\mathrm{pr})}{\to }{\mathcal{H}}_G^m(EG\times X) \\ \stackrel{{\left({\mathrm{ind}}_{G\to \{1\}}\right)}^{-1}}{\to }{\mathcal{H}}^m(EG{\times }_{G}X)\stackrel{{\mathcal{H}}^m(j)}{\to }{\mathcal{H}}^m\left((EG{\times }_{G}X{)}_{(n-1)}\right), \end{gathered}$$

$$\begin{gathered} {\lambda }_G^m(f:X\to L):{\left\{{\mathcal{H}}_G^m(X)/{\mathbb{I}}_G(L{)}^n\cdot {\mathcal{H}}_G^m(X)\right\}}_{n\ge 1} \\ \to {\left\{{\mathcal{H}}_G^m\left((EG{\times }_{G}X{)}_{(n-1)}\right)\right\}}_{n\ge 1}. \end{gathered}$$

$${\mathrm{invlim}}_{n\to \infty }^1{\mathcal{H}}^m\left((EG{\times }_{G}X{)}_{(n-1)}\right)$$

$${\mathrm{invlim}}_{n\to \infty }{\mathcal{H}}^m\left((EG{\times }_{G}X{)}_{(n-1)}\right)$$

$$\begin{gathered} 0\to {\mathrm{invlim}}_{n\to \infty }^1{\mathcal{H}}^{m-1}\left((EG{\times }_{G}X{)}_{(n-1)}\right)\to {\mathcal{H}}^m(EG{\times }_{G}X) \\ \to {\mathrm{invlim}}_{n\to \infty }{\mathcal{H}}^m\left((EG{\times }_{G}X{)}_{(n-1)}\right)\to 0 \end{gathered}$$

$${\mathcal{H}}^m(EG{\times }_{G}X)\cong {\mathrm{invlim}}_{n\to \infty }{\mathcal{H}}_G^m(X)/{\mathbb{I}}_G(L{)}^n\cdot {\mathcal{H}}_G^m(X).$$

$${\mathcal{H}}^m(BG)\cong {\mathrm{invlim}}_{n\to \infty }{\mathcal{H}}_G^m(\underline{E}G)/{\mathbb{I}}_G(\underline{E}G{)}^n\cdot {\mathcal{H}}_G^m(\underline{E}G).$$

$${\mathrm{ind}}_{G\to \{1\}}:{\mathcal{H}}^m(G\backslash X)\stackrel{\cong }{\to }{\mathcal{H}}_G^m(X).$$

$$\{{\mathcal{H}}^m(G\backslash X){\}}_{n\ge 1}\stackrel{\cong }{\to }\{{\mathcal{H}}^m(G\backslash X)/{\mathbb{I}}_G(L{)}^n\cdot {\mathcal{H}}^m(G\backslash X){\}}_{n\ge 1}$$

$${\left\{{\mathcal{H}}^m\left((G\backslash X{)}_{(n-1)}\right)\right\}}_{n\ge 1}\stackrel{\cong }{\to }{\left\{{\mathcal{H}}^m\left({\left(EG{\times }_{G}X\right)}_{(n-1)}\right)\right\}}_{n\ge 1}.$$

$$\{{\mathcal{H}}^m(G\backslash X){\}}_{n\ge 1}\stackrel{\cong }{\to }{\left\{{\mathcal{H}}^m\left((G\backslash X{)}_{(n-1)}\right)\right\}}_{n\ge 1}$$

$$\mathbb{I}(H)=\ker \left({\mathcal{H}}_H^0(\ast )\stackrel{{\mathcal{H}}_H^0(\mathrm{pr})}{\to }{\mathcal{H}}_H^0(H)\stackrel{{\mathrm{ind}}_{\{1\}\to H}}{\to }{\mathcal{H}}^0(\ast )\right)$$

$${\lambda }_H^m(\ast ):{\left\{{\mathcal{H}}_H^m(\ast )/\mathbb{I}(H{)}^n\right\}}_{n\ge 1}$$

Full text

0. Introduction

We first recall the Segal Conjecture for a finite group $G$. The equivariant stable cohomotopy groups $\pi^{n}_{G}(X)$ of a $G$-$CW$-complex are modules over the ring $\pi^{0}_{G}=\pi^{0}_{G}(\ast)$ which can be identified with the Burnside ring $A(G)$. Here and elsewhere $\ast$ denotes the one-point-space The augmentation homomorphism $A(G)\to{\mathbb{Z}}$ is the ring homomorphism sending the class of a finite set to its cardinality. The augmentation ideal ${\mathbb{I}}_{G}\subseteq A(G)$ is its kernel. Let $\pi^{m}_{G}(X)\widehat{{}_{{\mathbb{I}}_{G}}}$ be the ${\mathbb{I}}_{G}$-adic completion ${\operatorname{invlim}}_{n\to\infty}\pi^{m}_{G}(X)/{\mathbb{I}}_{G}^{n}\cdot\pi^{m}_{G}(X)$ of $\pi^{m}_{G}(X)$.

The following result was formulated as a conjecture by Segal and proved by Carlsson [6].

Theorem 0.1** (Segal Conjecture for finite groups).**

For every finite group $G$ and finite $G$-$CW$-complex $X$ there is an isomorphism

[TABLE]

In particular we get for $X=\ast$ and $m=0$ an isomorphism

[TABLE]

The purpose of this paper is to formulate and prove a version of it for infinite (discrete) groups, i.e., we will show

Theorem 0.2**.**

(Segal Conjecture for infinite groups). Let $G$ be a (discrete) group. Let $X$ be a finite proper $G$-$CW$-complex. Let $L$ be a proper finite dimensional $G$-$CW$-complex with the property that there is an upper bound on the order of its isotropy groups. Let $f\colon X\to L$ be a $G$-map.

Then the map of pro-${\mathbb{Z}}$-modules

[TABLE]

defined in (3.3) is an isomorphism of pro-${\mathbb{Z}}$-modules.

In particular we obtain an isomorphism

[TABLE]

If there is a finite $G$-$CW$-model for $\underline{E}G$, we obtain an isomorphism

[TABLE]

Here $\underline{E}G$ is the classifying space for proper $G$-actions and $\pi^{*}_{G}(X)$ is equivariant stable cohomotopy as defined in [10, Section 6]. The ideal ${\mathbb{I}}_{G}(L)$ is the augmentation ideal in the ring $\pi^{0}_{G}(L)$ (see Definition 3.1). We view $\pi^{*}_{G}(X)$ as $\pi^{0}_{G}(L)$-module by the multiplicative structure on equivariant stable cohomotopy and the map $f$. We denote by $\pi^{m}_{G}(X)\widehat{{}_{{\mathbb{I}}_{G}(L)}}$ its ${\mathbb{I}}_{G}(L)$-completion. More explanations will follow in the main body of the text.

In [10] various mutually distinct notions of a Burnside ring of a group $G$ are introduced which all agree with the standard notion for finite $G$. If there is a finite $G$-$CW$-model for $\underline{E}G$, then the homotopy theoretic definition is $A(G):=\pi^{0}_{G}(\underline{E}G)$, we define the ideal ${\mathbf{I}}_{G}\subseteq A(G)$ to be ${\mathbf{I}}_{G}(\underline{E}G)$, and we get in this notation from Theorem 0.2 an isomorphism

[TABLE]

We will actually formulate for every equivariant cohomology theory $\mathcal{H}_{?}^{*}$ with multiplicative structure a Completion Theorem (see Problem 3.4). It is not expected to be true in all cases. We give a strategy for its proof in Theorem 4.1. We show that this applies to equivariant stable cohomotopy, thus proving Theorem 0.2. It also applies to equivariant topological $K$-theory, where the Completion Theorem for infinite groups has already been proved in [15].

If $G$ is finite, we can take $L=\underline{E}G=\ast$ and then Theorem 0.2 reduces to Theorem 0.1. We will not give a new proof of Theorem 0.1, but use it as input in the proof of Theorem 0.2.

This paper is part of a general program to systematically study equivariant homotopy theory, which is well-established for finite groups and compact Lie groups, for infinite groups and non-compact Lie groups. The motivation comes among other things from the Baum-Connes Conjecture and the Farrell-Jones Conjecture.

Acknowledgement

The paper has been financially supported by the Leibniz-Preis of the author granted by the DFG, the ERC Advanced Grant “KL2MG-interactions” (no. 662400) of the author granted by the European Research Council, and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – GZ 2047/1, Projekt-ID 390685813.by the Cluster of Excellence “Hausdorff Center for Mathematics” at Bonn. The author wants to thank the referee for his detailed and useful report.

1. Equivariant Cohomology Theories with Multiplicative Structure

We briefly recall the axioms of a (proper) equivariant cohomology $\mathcal{H}_{?}^{*}$ theory with values in $R$-modules with multiplicative structure. More details can be found in [11].

Let $G$ be a (discrete) group. Let $R$ be a commutative ring with unit. A (proper) $G$-cohomology theory $\mathcal{H}^{*}_{G}$ with values in $R$-modules assigns to any pair $(X,A)$ of (proper) $G$-$CW$-complexes $(X,A)$ a ${\mathbb{Z}}$-graded $R$-module $\{\mathcal{H}^{n}_{G}(X,A)\mid n\in{\mathbb{Z}}\}$ such that $G$-homotopy invariance holds and there exists long exact sequences of pairs and long exact Mayer-Vietoris sequences. Often one also requires the disjoint union axiom what we will need not here since all our disjoint unions will be over finite index sets.

A multiplicative structure is given by a collection of $R$-bilinear pairings

[TABLE]

This product is required to be graded commutative, to be associative, to have a unit $1\in\mathcal{H}^{0}_{G}(X)$ for every (proper) $G$-$CW$-complex $X$, to be compatible with boundary homomorphisms and to be natural with respect to $G$-maps.

Let $\alpha\colon H\to G$ be a group homomorphism. Given an $H$-space $X$, define the induction of $X$ with $\alpha$ to be the $G$-space $\operatorname{ind}_{\alpha}X$ which is the quotient of $G\times X$ by the right $H$-action $(g,x)\cdot h:=(g\alpha(h),h^{-1}x)$ for $h\in H$ and $(g,x)\in G\times X$. If $\alpha\colon H\to G$ is an inclusion, we also write $\operatorname{ind}_{H}^{G}$ instead of $\operatorname{ind}_{\alpha}$.

A (proper) equivariant cohomology theory $\mathcal{H}_{?}^{*}$ with values in $R$-modules consists of a collection of (proper) $G$-cohomology theories $\mathcal{H}^{*}_{G}$ with values in $R$-modules for each group $G$ together with the following so called induction structure: given a group homomorphism $\alpha\colon H\to G$ and a (proper) $H$-$CW$-pair $(X,A)$ there are for each $n\in{\mathbb{Z}}$ natural homomorphisms

[TABLE]

If $\ker(\alpha)$ acts freely on $X$, then $\operatorname{ind}_{\alpha}\colon\mathcal{H}^{n}_{G}(\operatorname{ind}_{\alpha}(X,A))\to\mathcal{H}^{n}_{H}(X,A)$ is bijective for all $n\in{\mathbb{Z}}$. The induction structure is required to be compatibility with the boundary homomorphisms, to be functorial in $\alpha$ and to be compatible with inner automorphisms.

Sometimes we will need the following lemma whose elementary proof is analogous to the one in [9, Lemma 1.2].

Lemma 1.2**.**

Consider finite subgroups $H,K\subseteq G$ and an element $g\in G$ with $gHg^{-1}\subseteq K$. Let $R_{g^{-1}}\colon G/H\to G/K$ be the $G$-map sending $g^{\prime}H$ to $g^{\prime}g^{-1}K$ and $c(g)\colon H\to K$ be the group homomorphism sending $h$ to $ghg^{-1}$. Denote by $\operatorname{pr}\colon(\operatorname{ind}_{c(g)\colon H\to K}\ast)\to\ast$ the projection to the one-point space $\ast$.

Then the following diagram commutes

[TABLE]

Let $\mathcal{H}_{?}^{*}$ be a (proper) equivariant cohomology theory. A multiplicative structure on it assigns a multiplicative structure to the associated (proper) $G$-cohomology theory $\mathcal{H}^{*}_{G}$ for every group $G$ such that for each group homomorphism $\alpha\colon H\to G$ the maps given by the induction structure of (1.1) are compatible with the multiplicative structures on $\mathcal{H}^{*}_{G}$ and $\mathcal{H}^{*}_{H}$.

Example 1.3**.**

Equivariant cohomology theories coming from non-equivariant ones. Let $\mathcal{K}^{*}$ be a (non-equivariant) cohomology theory with multiplicative structure, for instance singular cohomology or topological $K$-theory. We can assign to it an equivariant cohomology theory with multiplicative structure $\mathcal{H}_{?}^{*}$ in two ways. Namely, for a group $G$ and a pair of $G$-$CW$-complexes $(X,A)$ we define $\mathcal{H}^{n}_{G}(X,A)$ by $\mathcal{K}^{n}(G\backslash(X,A))$ or by $\mathcal{K}^{n}(EG\times_{G}(X,A))$.

Example 1.4**.**

(Equivariant topological $K$-theory).

In [15] equivariant topological $K$-theory is defined for finite proper equivariant $CW$-complexes in terms of equivariant vector bundles. It reduces to the classical notion which appears for instance in [2]. Its relation to equivariant $KK$-theory is explained in [16]. This definition is extended to (not necessarily finite) proper equivariant $CW$-complexes in [15] in terms of equivariant spectra using $\Gamma$-spaces and yields a proper equivariant cohomology theory $K_{?}^{*}$ with multiplicative as explained in [11, Example 1.7]. It has the property that for any finite subgroup $H$ of a group $G$ we have

[TABLE]

where $R_{{\mathbb{C}}}(H)$ denote the complex representation ring of $H$.

Example 1.5**.**

(Equivariant Stable Cohomotopy).

In [10, Section 6] equivariant stable cohomotopy $\pi_{?}^{*}$ is defined for finite proper equivariant $CW$-complexes in terms of maps of sphere bundles associated of equivariant vector bundles. For finite $G$ it reduces to the classical notion. This definition is extended to arbitrary proper $G$-$CW$-complexes by Degrijse-Hausmann-Lueck-Patchkoria-Schwede [7], where a systematic study of equivariant homotopy theory for (not necessarily compact) Lie groups and proper $G$-$CW$-complexes is developed.

Let $H\subseteq G$ be a finite subgroup. Recall that by the induction structure we have $\pi^{n}_{G}(G/H)=\pi^{n}_{H}(\ast)$. The equivariant stable homotopy groups $\pi_{H}^{n}$ are computed in terms of the splitting due to Segal and tom Dieck (see [17, Proposition 2] and [18, Theorem 7.7 in Chapter II on page 154]) by

[TABLE]

where $\pi_{-n}^{s}$ denotes (non-equivariant) stable homotopy and $(K)$ runs through the conjugacy classes of subgroups of $H$. In particular we get

[TABLE]

where $A(H)$ is the Burnside ring.

2. Some Preliminaries about Pro-Modules

It will be crucial to handle pro-systems and pro-isomorphisms and not to pass directly to inverse limits. In this section we fix our notation for handling pro-$R$-modules for a commutative ring with unit $R$. For the definitions in full generality see for instance [1, Appendix] or [4, §2].

For simplicity, all pro-$R$-modules dealt with here will be indexed by the positive integers. We write $\{M_{n},\alpha_{n}\}$ or briefly $\{M_{n}\}$ for the inverse system

[TABLE]

and also write $\alpha_{n}^{m}:=\alpha_{m+1}\circ\cdots\circ\alpha_{n}\colon M_{n}\to M_{m}$ for $n>m$ and put $\alpha^{n}_{n}=\operatorname{id}_{M_{n}}$. For the purposes here, it will suffice (and greatly simplify the notation) to work with “strict” pro-homomorphisms $\{f_{n}\}\colon\{M_{n},\alpha_{n}\}\to\{N_{n},\beta_{n}\}$, i.e., a collection of homomorphisms $f_{n}\colon M_{n}\to N_{n}$ for $n\geq 1$ such that $\beta_{n}\circ f_{n}=f_{n-1}\circ\alpha_{n}$ holds for each $n\geq 2$. Kernels and cokernels of strict homomorphisms are defined in the obvious way, namely levelwise.

A pro-$R$-module $\{M_{n},\alpha_{n}\}$ will be called pro-trivial if for each $m\geq 1$, there is some $n\geq m$ such that $\alpha^{m}_{n}=0$. A strict homomorphism $f\colon\{M_{n},\alpha_{n}\}\to\{N_{n},\beta_{n}\}$ is a pro-isomorphism if and only if $\ker(f)$ and $\operatorname{coker}(f)$ are both pro-trivial, or, equivalently, for each $m\geq 1$ there is some $n\geq m$ such that $\operatorname{im}(\beta_{n}^{m})\subseteq\operatorname{im}(f_{m})$ and $\ker(f_{n})\subseteq\ker(\alpha_{n}^{m})$. A sequence of strict homomorphisms

[TABLE]

will be called exact if the sequences of $R$-modules $M_{n}\xrightarrow{f_{n}}N_{n}\xrightarrow{g_{n}}M_{n}^{\prime\prime}$ is exact for each $n\geq 1$, and it is called pro-exact if $g_{n}\circ f_{n}=0$ holds for $n\geq 1$ and the pro-$R$-module \{\ker(g_{n})/\operatorname{im}(f_{n})\bigr{\}} is pro-trivial.

The elementary proofs of the following two lemmas can be found for instance in [13, Section 2].

Lemma 2.1**.**

Let $0\to\{M_{n}^{\prime},\alpha_{n}^{\prime}\}\xrightarrow{\{f_{n}\}}\{M_{n},\alpha_{n}\}\xrightarrow{\{g_{n}\}}\{M_{n}^{\prime\prime},\alpha_{n}^{\prime\prime}\}\to 0$ be a pro-exact sequence of pro-$R$-modules. Then there is a natural exact sequence

[TABLE]

In particular a pro-isomorphism $\{f_{n}\}\colon\{M_{n},\alpha_{n}\}\to\{N_{n},\beta_{n}\}$ induces isomorphisms

[TABLE]

Lemma 2.2**.**

Fix any commutative Noetherian ring $R$, and any ideal $I\subseteq R$. Then for any exact sequence $M^{\prime}\to M\to M^{\prime\prime}$ of finitely generated $R$-modules, the sequence

[TABLE]

of pro-$R$-modules is pro-exact.

3. The Formulation of a Completion Theorem

Consider a proper equivariant $G$-cohomology theory $\mathcal{H}_{?}^{*}$ with multiplicative structure. In the sequel $\mathcal{H}^{*}$ is the non-equivariant cohomology theory with multiplicative structure given by $\mathcal{H}^{*}_{G}$ for $G=\{1\}$. Notice that $\mathcal{H}^{0}(\ast)$ is a commutative ring with unit and $\mathcal{H}^{n}_{G}(X)$ is a $\mathcal{H}^{0}(\ast)$-module. In some future applications $\mathcal{H}^{0}(\ast)$ will be just ${\mathbb{Z}}$. In the sequel $[Y,X]^{G}$ denotes the set of $G$-homotopy classes of $G$-maps $Y\to X$. Notice that evaluation at the unit element of $G$ induces a bijection $[G,X]^{G}\xrightarrow{\cong}\pi_{0}(X)$. It is compatible with the left $G$-actions, which are on the source induced by precomposing with right multiplication $r_{g}\colon G\to G,g^{\prime}\mapsto g^{\prime}g$ and on the target by the given left $G$-action on $X$.

So we can represent elements in $G\backslash\pi_{0}(X)$ by classes $\overline{x}$ of $G$-maps $x\colon G\to X$, where $x\colon G\to X$ and $y\colon G\to X$ are equivalent, if for some $g\in G$ the composite $y\circ r_{g}$ is $G$-homotopic to $x$.

Definition 3.1** (Augmentation ideal).**

For any proper $G$-CW-complex $X$ the augmentation module ${\mathbb{I}}_{G}^{n}(X)\subseteq\mathcal{H}^{n}_{G}(X)$ is defined as the kernel of the map

[TABLE]

(The composite above is independent of the choice of $x\in\overline{x}$ by $G$-homotopy invariance and Lemma 1.2.) If $n=0$, the map above is a ring homomorphism and ${\mathbb{I}}_{G}(X):={\mathbb{I}}^{0}_{G}(X)$ is an ideal called the augmentation ideal.

Given a $G$-map $f\colon X\to Y$, the induced map $\mathcal{H}^{n}_{G}(f)\colon\mathcal{H}^{n}_{G}(Y)\to\mathcal{H}^{n}_{G}(X)$ restricts to a map ${\mathbb{I}}_{G}^{n}(Y)\to{\mathbb{I}}_{G}^{n}(X)$.

We will need the following elementary lemma:

Lemma 3.2**.**

Let $X$ be a CW-complex of dimension $(n-1)$. Then any $n$-fold product of elements in ${\mathbb{I}}_{G}^{*}(X)$ is zero.

Proof.

Write $X=Y\cup A$, where $Y$ and $A$ are closed subsets, $Y$ contains $X^{(n-2)}$ as a homotopy deformation retract, and $A$ is a disjoint union of $(n{-}1)$-disks. Fix elements $v_{1},v_{2},\dots,v_{n}\in{\mathbb{I}}_{G}^{*}(X)$. We can assume by induction that $v_{1}\cdots{}v_{n-1}$ vanishes after restricting to $Y$, and hence that it is the image of an element $u\in\mathcal{H}_{G}^{*}(X,Y)$. Also, $v_{n}$ clearly vanishes after restricting to $A$, and hence is the image of an element $v\in\mathcal{H}_{G}^{*}(X,A)$. The product of $v_{1}\cdots{}v_{n-1}$ and $v_{n}$ is the image in $\mathcal{H}_{G}^{*}(X)$ of the element $uv\in\mathcal{H}_{G}^{*}(X,Y\cup A)=\mathcal{H}_{G}^{*}(X,X)=0$ and so $v_{1}\cdots{}v_{n}=0$. ∎

Now fix a map $f\colon X\to L$ between $G$-$CW$-complexes. Consider $\mathcal{H}^{*}_{G}(X)$ as a module over the ring $\mathcal{H}^{0}_{G}(L)$. Consider the composition

[TABLE]

where $i$ and $j$ denote the inclusions, $\operatorname{pr}$ the projection and $(EG\times_{G}X)_{(n-1)}$ is the $(n-1)$-skeleton of $EG\times_{G}X$. This composite is zero because of Lemma 3.2 since its image is contained in ${\mathbb{I}}^{n}\left((EG\times_{G}X)_{(n-1)}\right)$. Thus we obtain a homomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules

[TABLE]

We will sometimes write $\lambda^{m}_{G}$ or $\lambda^{m}_{G}(X)$ instead of $\lambda^{m}_{G}(f\colon X\to L)$ if the map $f\colon X\to L$ is clear from the context. Notice that the target of $\lambda^{m}_{G}(f\colon X\to L)$ depends only on $X$ but not on the map $f\colon X\to L$, whereas the source does depend on $f$.

Problem 3.4** (Completion Problem).**

Under which conditions on ${\mathcal{H}^{*}_{?}}$ and $L$ is the map of pro-$\mathcal{H}^{0}(\ast)$-modules $\lambda^{m}_{G}(f\colon X\to L)$ defined in (3.3) an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules?

Remark 3.5** (Consequences of the Completion Theorem).**

Suppose that the map of pro-$\mathcal{H}^{0}(\ast)$-modules $\lambda^{m}_{G}(X)$ defined in (3.3) is an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules. Obviously the pro-module $\{\mathcal{H}^{m}_{G}(X)/{\mathbb{I}}_{G}(L)^{n}\cdot\mathcal{H}^{m}_{G}(X)\}_{n\geq 1}$ satisfies the Mittag-Leffler condition since all structure maps are surjective. This implies that its ${\lim}^{1}$-term vanishes. We conclude from Lemma 2.1

[TABLE]

Milnor’s exact sequence

[TABLE]

implies that we obtain an isomorphism

[TABLE]

Remark 3.6** (Taking $L=\underline{E}G$).**

The classifying space $\underline{E}G$ for proper $G$-actions is a proper $G$-$CW$-complex such that the $H$-fixed point set is contractible for every finite subgroup $H\subseteq G$. It has the universal property that for every proper $G$-$CW$-complex $X$ there is up to $G$-homotopy precisely one $G$-map $f\colon X\to\underline{E}G$. Recall that a $G$-$CW$-complex is proper if and only if all its isotropy groups are finite and is finite if and only if it is cocompact. There is a cocompact $G$-$CW$-model for the classifying space $\underline{E}G$ for proper $G$-actions for instance if $G$ is word-hyperbolic in the sense of Gromov, if $G$ is a cocompact subgroup of a Lie group with finitely many path components, if $G$ is a finitely generated one-relator group, if $G$ is an arithmetic group, a mapping class group of a compact surface or the group of outer automorphisms of a finitely generated free group. For more information about $\underline{E}G$ we refer for instance to [5] and [12].

Suppose that there is a finite model for the classifying space of proper $G$-actions $\underline{E}G$. Then we can apply this to $\operatorname{id}\colon\underline{E}G\to\underline{E}G$ and obtain an isomorphism

[TABLE]

Remark 3.7** (The free case).**

The statement of the Completion Theorem as stated in Problem 3.4 is always true for trivial reasons if $X$ is a free finite $G$-$CW$-complex. Then induction induces an isomorphism

[TABLE]

Since ${\mathbb{I}}(G\backslash X)^{n}=0$ for large enough $n$ by Lemma 3.2, the canonical map

[TABLE]

with the constant pro-$\mathcal{H}^{0}(\ast)$-module as source is an isomorphism. Hence the source of $\lambda^{m}_{G}(f\colon G\to X)$ can be identified with constant pro-$\mathcal{H}^{0}(\ast)$-module $\{\mathcal{H}^{m}(G\backslash X)\}_{n\geq 1}$.

The projection $EG\times_{G}X\to G\backslash X$ is a homotopy equivalence and induces an isomorphism pro-${\mathbb{Z}}$-modules

[TABLE]

Since $G\backslash X$ is finite dimensional, the canonical map

[TABLE]

is an isomorphism of pro-${\mathbb{Z}}$-modules. Hence also the target of $\lambda^{m}_{G}(f\colon G\to X)$ can be identified with constant pro-$\mathcal{H}^{0}(\ast)$-module $\{\mathcal{H}^{m}(G\backslash X)\}_{n\geq 1}$. One easily checks that under these identifications $\lambda^{m}_{G}(f\colon G\to X)$ is the identity.

Hence the Completion Theorem is only interesting in the case, where $G$ contains torsion.

4. A Strategy for a Proof of a Completion Theorem

Theorem 4.1**.**

(Strategy for the proof of Theorem 0.2).* Let $\mathcal{H}^{?}_{*}$ be an equivariant cohomology theory with values in $R$-modules with a multiplicative structure. Let $L$ be a proper $G$-$CW$-complex. Suppose that the following conditions are satisfied, where $\mathcal{F}(L)$ is the family of subgroups of $G$ given by $\{H\subseteq G\mid L^{H}\not=\emptyset\}$.*

1. (1)

The ring $\mathcal{H}^{0}(\ast)$ is Noetherian; 2. (2)

For any $H\in\mathcal{F}(L)$ and $m\in{\mathbb{Z}}$ the $\mathcal{H}^{0}(\ast)$-module $\mathcal{H}^{m}_{H}(\ast)$ is finitely generated; 3. (3)

Let $H\in\mathcal{F}(L)$, let $\mathcal{P}\subseteq\mathcal{H}^{0}_{H}(\ast)$ be a prime ideal, and let $f\colon G/H\to L$ be any $G$-map. Then the augmentation ideal

[TABLE]

is contained in $\mathcal{P}$ if $\mathcal{H}^{0}_{G}(L)\xrightarrow{\mathcal{H}^{0}_{G}(f)}\mathcal{H}^{0}_{G}(G/H)\xrightarrow{\operatorname{ind}_{G\to\{1\}}}\mathcal{H}^{0}_{H}(\ast)$ maps ${\mathbb{I}}_{G}(L)$ into $\mathcal{P}$; 4. (4)

The Completion Theorem is true for every finite group $H$ with $H\in\mathcal{F}(L)$ in the case, where $X=L=\ast$ and $f=\operatorname{id}\colon\ast\to\ast$. In other words, for every finite group $H$ with $L^{H}\not=\emptyset$ the map of pro-$\mathcal{H}^{0}(\ast)$-modules

[TABLE]

defined in (3.3) is an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules.

Then, under the conditions above, the Completion Theorem is true for $\mathcal{H}^{?}_{*}$ and every $G$-map $f\colon X\to L$ from a finite proper $G$-$CW$-complex $X$ to $L$, i.e., the map of pro-$\mathcal{H}^{0}(\ast)$-modules

[TABLE]

defined in (3.3) is an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules.

Proof.

We first prove the Completion Theorem for $X=G/H$, i.e., for any a $G$-map $f\colon G/H\to L$. Obviously $H$ belongs to $\mathcal{F}(L)$. The following diagram of pro-modules commutes

[TABLE]

where $\operatorname{pr}$ denotes the obvious projection. The lower horizontal arrow is an isomorphism of pro-modules by condition (4). The right vertical arrow and the upper left vertical arrow are obviously isomorphisms of pro-modules. Hence the upper horizontal arrow is an isomorphism of pro-modules if we can show that the lower left vertical arrow is an isomorphism of pro-modules.

Let $I_{f}$ be the image of ${\mathbb{I}}_{G}(L)$ under the composite of ring homomorphisms

[TABLE]

Let $J_{f}$ be the ideal in $\mathcal{H}^{0}_{H}(\ast)$ generated by $I_{f}$. Obviously $I_{f}\subseteq J_{f}\subseteq{\mathbb{I}}(H)$. Then the left lower vertical arrow is the composite

[TABLE]

where the first map is already levelwise an isomorphisms and in particular an isomorphism of pro-modules. In order to show that the second map is an isomorphism of pro-modules, it remains to show that ${\mathbb{I}}(H)^{k}\subseteq J_{f}$ for an appropriate integer $k\geq 1$. Equivalently, we want to show that the ideal ${\mathbb{I}}(H)/J_{f}$ of the quotient ring $\mathcal{H}^{0}_{H}(\ast)/J_{f}$ is nilpotent. Since $\mathcal{H}^{0}_{H}(\ast)$ is Noetherian by conditions (1) and (2), the ideal ${\mathbb{I}}(H)/J_{f}$ is finitely generated. Hence it suffices to show that ${\mathbb{I}}(H)/J_{f}$ is contained in the nilradical, i.e., the ideal consisting of all nilpotent elements, of $\mathcal{H}^{0}_{H}(\ast)/J_{f}$. The nilradical agrees with the intersection of all the prime ideals of $\mathcal{H}^{0}_{H}(\ast)/J_{f}$ by [3, Proposition 1.8]. The preimage of a prime ideal in $\mathcal{H}^{0}_{H}(\ast)/J_{f}$ under the projection $\mathcal{H}^{0}_{H}(\ast)\to\mathcal{H}^{0}_{H}(\ast)/J_{f}$ is again a prime ideal. Hence it remains to show that any prime ideal of $\mathcal{H}^{0}_{H}(\ast)$ which contains $I_{f}$ also contains ${\mathbb{I}}(H)$. But this is guaranteed by condition (3). This finishes the proof in the case $X=G/H$.

The general case of a $G$-map $f\colon X\to L$ from a finite $G$-$CW$-complex $X$ to a $G$-$CW$-complex $L$ is done by induction over the dimension $r$ of $X$ and subinduction over the number of top-dimensional equivariant cells. For the induction step we write $X$ as a $G$-pushout

[TABLE]

In the sequel we equip $G/H\times S^{r-1}$, $Y$ and $G/H\times D^{r}$ with the maps to $L$ given by the composite of $f\colon X\to L$ with $k\circ q$, $k$ and $Q$. The long exact Mayer-Vietoris sequence of the $G$-pushout above is a long exact sequence of $\mathcal{H}^{0}_{G}(L)$-modules and looks like

[TABLE]

Condition (2) implies that $\mathcal{H}^{m}_{G}(G/H)$ and $\mathcal{H}^{m}_{G}(G/H\times D^{r})$ are finitely generated as $\mathcal{H}^{0}(\ast)$-modules. Since $\mathcal{H}^{0}(\ast)$ is Noetherian by condition (1), the $\mathcal{H}^{0}(\ast)$-module $\mathcal{H}^{m}_{G}(X)$ is finitely generated provided that the $\mathcal{H}^{0}(\ast)$-module $\mathcal{H}^{m}_{G}(Y)$ is finitely generated. Thus we can show inductively that the $\mathcal{H}^{0}(\ast)$-module $\mathcal{H}^{m}_{G}(X)$ is finitely generated for every $m\in{\mathbb{Z}}$. In particular the ring $\mathcal{H}^{0}_{G}(X)$ is Noetherian. Let $J\subseteq\mathcal{H}^{0}_{G}(X)$ be the ideal generated by the image of ${\mathbb{I}}_{G}(L)$ under the ring homomorphism $\mathcal{H}^{0}_{G}(L)\to\mathcal{H}^{0}_{G}(X)$. Then for every $\mathcal{H}^{0}_{G}(X)$-module the obvious map $\{M/{\mathbb{I}}_{G}(L)^{n}\cdot M\}_{n\geq 1}\to\{M/J^{n}\cdot M\}_{n\geq 1}$ is levelwise an isomorphism and in particular an isomorphism of $\mathcal{H}^{0}_{G}(X)$-modules. We conclude from Lemma 2.2 that the following sequence of pro-$\mathcal{H}^{0}(\ast)$-modules is exact, where $M/I$ stands for $M/I\cdot M$.

[TABLE]

Applying $EG_{(n-1)}\times_{G}-$ to the $G$-pushout above yields a pushout and thus a long exact Mayer-Vietoris sequence

[TABLE]

The obvious map

[TABLE]

is an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules for any finite dimensional $G$-$CW$-complex $Z$. Hence we obtain a long exact sequence of pro-$\mathcal{H}^{0}(\ast)$-modules

[TABLE]

Now the various maps $\lambda^{m}_{G}$ induce a map from the long exact sequence of pro-$\mathcal{H}^{0}(\ast)$-modules (4.2) to the long exact sequence of pro-$\mathcal{H}^{0}(\ast)$-modules (4.3). The maps for $G/H\times S^{r-1}$, $G/H\times D^{r}$ and $Y$ are isomorphisms of pro-$\mathcal{H}^{0}(\ast)$-modules by induction hypothesis and by $G$-homotopy invariance applied to the $G$-homotopy equivalence $G/H\times D^{r}\to G/H$. By the Five-Lemma for maps of pro-modules the map

[TABLE]

is an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules. This finishes the proof of Theorem 4.1. ∎

The next lemma will be needed to check condition (3) appearing in Theorem 4.1.

Given an G-cohomology theory $\mathcal{H}^{*}_{G}$. there is an equivariant version of the Atiyah-Hirzebruch spectral sequence of $\mathcal{H}^{0}(\ast)$-modules which converges to $\mathcal{H}^{p+q}(L)$ in the usual sense provided that $L$ is finite dimensional, and whose $E_{2}$-term is

[TABLE]

where $H^{p}_{G}(X;\mathcal{H}^{q}_{G}(G/?))$ is the Bredon cohomology of $L$ with coefficients in the ${\mathbb{Z}}\EuR{Or}(G)$-module sending $G/H$ to $\mathcal{H}^{q}_{G}(G/H)$. If $\mathcal{H}^{*}_{G}$ comes with a multiplicative structure, then this spectral sequence comes with a multiplicative structure.

Lemma 4.4**.**

Suppose that $L$ is a $l$-dimensional proper $G$-$CW$-complex for some positive integer $l$. Suppose that for $r=2,3,\ldots,l$ the differential appearing in the Atiyah-Hirzebruch spectral sequence for $L$ and $\mathcal{H}^{*}_{G}$

[TABLE]

vanishes rationally.

1. (1)

Then we can find for a given $x\in H^{0}_{G}(L;\mathcal{H}^{0}_{G}(G/?))$ a positive integer $k$ such that $x^{k}$ is contained in the image of the edge homomorphism

[TABLE] 2. (2)

Let $H\in\mathcal{F}(L)$, let $\mathcal{P}\subseteq\mathcal{H}^{0}_{H}(\ast)$ be a prime ideal and let $f\colon G/H\to L$ be any $G$-map. Suppose that the augmentation ideal

[TABLE]

is contained in $\mathcal{P}$ if $\mathcal{P}$ contains the image of the structure map for $H$ of the inverse limit over the orbit category $\EuR{Or}(G;\mathcal{F}(L))$ associated to the family $\mathcal{F}(L)$

[TABLE]

Then condition (3) appearing in Theorem 4.1 is satisfied for $H$, $\mathcal{P}$ and $f$.

Proof.

(1) Consider $x\in H^{0}_{G}(L;\mathcal{H}^{0}_{G}(G/?))$. We construct inductively positive integers $k_{1}$, $k_{2}$, $\ldots,$ $k_{l}$ such that

[TABLE]

Put $k_{1}=1$. We have $H^{0}_{G}(L;\mathcal{H}^{0}_{G}(G/?))=E^{0,0}_{2}$ and hence $x=x^{1}=x^{\prod_{i=1}^{1}k_{i}}\in E^{0,0}_{2}$. This finishes the induction beginning $r=1$.

In the induction step from $(r-1)$ to $r\geq 2$ we can assume that we have already constructed $k_{1},\ldots,k_{r-1}$ and shown that $x^{\prod_{i=1}^{r-1}k_{i}}$ belongs to $E^{0,0}_{r}$. Now choose $k_{r}$ with $k_{r}\cdot d^{0,0}_{r}\left(x^{\prod_{i=1}^{r-1}k_{i}}\right)=0$. This is possible since by assumption $d^{0,0}_{r}\otimes_{{\mathbb{Z}}}\operatorname{id}_{{\mathbb{Q}}}=0$. For any element $y\in E^{0,0}_{r}$ one checks inductively for $j=1,2,\ldots$

[TABLE]

This implies

[TABLE]

Since $E^{0,0}_{r+1}$ is the kernel of $d^{0,0}_{r}\colon E^{0,0}_{r}\to E^{0,0}_{r+1}$, we conclude $x^{\prod_{i=1}^{r}k_{i}}\in E^{0,0}_{r+1}$. Since $L$ is $l$-dimensional, we get for $k=\prod_{i=1}^{l}k_{i}$ that $x^{k}\in E_{\infty}^{0,0}$. Since $E_{\infty}^{0,0}$ is the image of the edge homomorphism $\operatorname{edge}^{0,0}$, assertion 1 follows.

 (2) Consider the following commutative diagram

[TABLE]

Here $\alpha$ is the isomorphism, which sends $v\in H^{0}_{G}\left(E_{\mathcal{F}(L)}(G);\mathcal{H}^{0}_{G}(G/?)\right)$ to the system of elements that is for $G/K\in\EuR{Or}(G;\mathcal{F}(L))$ the image of $v$ under the homomorphism

[TABLE]

for the up to $G$-homotopy unique $G$-map $i_{K}\colon G/K\to E_{\mathcal{F}(L)}(G)$. The $G$-map $u\colon L\to E_{\mathcal{F}(L)}(G)$ is the up to $G$-homotopy unique $G$-map from $L$ to the classifying space of the family $\mathcal{F}(L)$, and $\Phi_{H}$ is the structure map of the inverse limit for $H$. We have to prove that ${\mathbb{I}}(H)$ is contained in the prime ideal $\mathcal{P}$ provided that $\mathcal{P}$ contains the image of ${\mathbb{I}}_{G}(L)$ under the composite $\mathcal{H}^{0}_{G}(L)\xrightarrow{\mathcal{H}^{0}_{G}(f)}\mathcal{H}^{0}_{G}(G/H)\xrightarrow{\operatorname{ind}_{H\to G}}\mathcal{H}^{0}_{H}(\ast)$.

Consider $a\in{\operatorname{invlim}}_{G/K\in\EuR{Or}(G;\mathcal{F}(L))}{\mathbb{I}}(K)$. Let $x\in H^{0}_{G}\left(L;\mathcal{H}^{0}_{G}(G/?)\right)$ be the image of $a$ under the composite

[TABLE]

We conclude from assertion (1) that for some positive number $k$ there is an element $y\in\mathcal{H}^{0}_{G}(L)$ with $\operatorname{edge}^{0,0}(y)=x^{k}$. One easily checks that $y$ belongs to ${\mathbb{I}}_{G}(L)$, just inspect the diagram above for $H=\{1\}$. Hence the composite

[TABLE]

maps $y$ to $\mathcal{P}$ by assumption. An easy diagram chase shows that

[TABLE]

maps $a^{k}$ to $\mathcal{P}$. Since $\mathcal{P}$ is a prime ideal and $\phi_{H}$ is multiplicative, $\phi_{H}$ sends $a$ to $\mathcal{P}$. Hence the image of $\phi_{H}\colon{\operatorname{invlim}}_{G/K\in\EuR{Or}(G;\mathcal{F}(L))}{\mathbb{I}}(K)\to{\mathbb{I}}(H)$ lies $\mathcal{P}$. Hence we get by assumption ${\mathbb{I}}(H)\subseteq\mathcal{P}$. This finishes the proof of Lemma 4.4. ∎

5. The Segal Conjecture for Infinite Groups

In this section we prove the Segal Conjecture 0.2 for infinite groups. It is just the Completion Theorem formulated in Problem 3.4 for equivariant stable cohomotopy $\mathcal{H}_{?}^{*}=\pi_{?}^{*}$ under the condition that there is an upper bound on the orders of finite subgroups on $L$ and $L$ has finite dimension.

Proof of Theorem 0.2.

We want to apply Theorem 4.1 and therefore have to prove conditions (1), (2), (3) and (4) appearing there.

Condition (1) appearing there is satisfied because of $\pi^{0}_{s}(\ast)={\mathbb{Z}}$.

Condition (2) is satisfied because of Example 1.5.

Next we prove condition (3). Recall the assumption that there is an upper bound on the orders of finite subgroups of $L$ and that $L$ is finite dimensional. Recall that $\mathcal{F}(L)$ denotes the family of finite subgroups $H\subseteq G$ with $L^{H}\not=\emptyset$. We can find by Example 1.5 for every $q\in{\mathbb{Z}}$ with $q\not=0$ a positive integer $C(q)$ such that the order of $\pi^{q}_{H}(\ast)$ divides $C(q)$ for every $H\in\mathcal{F}(L)$. Furthermore recall that $L$ is finite dimensional. Consider the equivariant cohomological Atiyah-Hirzebruch spectral sequence converging to $\pi^{p+q}_{G}(L)$. Its $E_{2}$-term is given by

[TABLE]

Therefore $E^{r,1-r}_{r}$ is annihilated by multiplication with $C(1-r)$ and hence rationally trivial for $r\geq 2$. Hence for $r\geq 2$ the differential

[TABLE]

vanishes rationally. We have shown that the conditions appearing in Lemma 4.4 are satisfied. Hence in order to verify condition (3), it suffices to prove for any family $\mathcal{F}$ of subgroups of $G$ with the property that there exists an upper bound on the orders of subgroups appearing $\mathcal{F}$, any $H\in\mathcal{F}$ and any prime ideal $\mathcal{P}$ of the Burnside ring $A(H)$ that $\mathcal{P}$ contains the augmentation ideal ${\mathbb{I}}_{H}$ provided $\mathcal{P}$ contains the image of the structure map for $H$ of the inverse limit

[TABLE]

Fix a finite group $H$. We begin with recalling some basics about the prime ideals in the Burnside ring $A(H)$ taken from [8]. In the sequel $p$ is a prime number or $p=0$. For a subgroup $K\subseteq H$ let $\mathcal{P}(K,p)$ be the preimage of $p\cdot{\mathbb{Z}}$ under the character map for $K$

[TABLE]

This is a prime ideal and each prime ideal of $A(H)$ is of the form $\mathcal{P}(K,p)$. If $\mathcal{P}(K,p)=\mathcal{P}(L,q)$, then $p=q$. If $p$ is a prime, then $\mathcal{P}(K,p)=\mathcal{P}(L,p)$ if and only if $(K[p])=(L[p])$, where $K[p]$ is the minimal normal subgroup of $K$ with a $p$-group as quotient. Notice for the sequel that $K[p]=\{1\}$ if and only if $K$ is a $p$-group. If $p=0$, then $\mathcal{P}(K,p)=\mathcal{P}(L,p)$ if and only if $(K)=(L)$.

Fix a prime ideal $\mathcal{P}=\mathcal{P}(K,p)$. Choose a positive integer $m$ such that $|H|$ divides $m$ for all $H\in\mathcal{F}$. Fix $H\in\mathcal{F}$. Choose a free $H$-set $S$ together with a bijection $u\colon S\xrightarrow{\cong}[m]$, where $[m]=\{1,2,\ldots,m\}$. Such $S$ exists since $|H|$ divides $m$ and we can take for $S$ the disjoint union of $\frac{m}{|H|}$ copies of $H$. Thus we obtain an injective group homomorphism

[TABLE]

where $l_{h}\colon S\to S$ is given by left multiplication with $h$ and $S_{m}=\operatorname{aut}([m])$ is the group of permutations of $[m]$. Let $S_{m}[\rho_{u}]$ denote the $H$-set obtained from $S_{m}$ by the $H$-action $h\cdot\sigma:=\rho_{u}(h)\circ\sigma$. Let $\operatorname{Syl}_{p}(S_{m})$ be the $p$-Sylow subgroup of $S_{m}$. Let $S_{m}/\operatorname{Syl}_{p}(S_{m})[\rho_{u}]$ denote the $H$-set obtained from the homogeneous space $S_{m}/\operatorname{Syl}_{p}(S_{m})$ by the $H$-action given by $h\cdot\overline{\sigma}=\overline{\rho_{u}(h)\circ\sigma}$. The $H$-action on $S_{m}[\rho_{u}]$ is free. If for $K\subseteq H$ we have $\left(S_{m}/\operatorname{Syl}_{p}(S_{m})[\rho_{u}]\right)^{K}\not=\emptyset$, then for some $\sigma\in S_{m}$ we get $\rho_{u}(K)\subseteq\sigma\cdot\operatorname{Syl}_{p}(S_{m})\cdot\sigma^{-1}$ and hence $K$ must be a $p$-group.

Suppose that $T$ is another free $H$-set together with a bijection $v\colon T\xrightarrow{\cong}[m]$. Then we can choose an $H$-isomorphism $w\colon S\to T$. Let $\tau\in S_{m}$ be given by the composition $v\circ w\circ u^{-1}$. Then $c(\tau)\circ\rho_{u}=\rho_{v}$ holds, where $c(\tau)\colon S_{m}\to S_{m}$ sends $\sigma$ to $\tau\circ\sigma\circ\tau^{-1}$. Moreover, left multiplication with $\tau$ induces isomorphisms of $H$-sets

[TABLE]

Hence we obtain elements in A(H)

[TABLE]

which are independent of the choice of $S$ and $u\colon S\xrightarrow{\cong}[m]$. If $i\colon H_{0}\to H_{1}$ is an injective group homomorphisms between elements in $\mathcal{F}$, then one easily checks that the restriction homomorphism $A(i)\colon A(H_{1})\to A(H_{0})$ sends $[S_{m}]$ to $[S_{m}]$ and $[S_{m}/\operatorname{Syl}_{p}(S_{m})]$ to $[S_{m}/\operatorname{Syl}_{p}(S_{m})]$. Thus we obtain elements

[TABLE]

Define elements

[TABLE]

by the collection of elements $|S_{m}|\cdot[K/K]$ and $|S_{m}/\operatorname{Syl}_{p}(S_{m})|\cdot[K/K]$ in $A(K)$ for $K\in\mathcal{F}$. Thus we get elements

[TABLE]

The image of $[[S_{m}]]-|S_{m}|\cdot 1$ and $[[S_{m}/\operatorname{Syl}_{p}(S_{m})]]-|S_{m}/\operatorname{Syl}_{p}(S_{m})|\cdot 1$ respectively under the structure map of the inverse limit ${\operatorname{invlim}}_{G/K\in\EuR{Or}(G;\mathcal{F})}{\mathbb{I}}_{K}$ for the object $G/H\in\EuR{Or}(G;\mathcal{F})$ is $[S_{m}]-|S_{m}|\cdot[H/H]$ and $[S_{m}/\operatorname{Syl}_{p}(S_{m})]-|S_{m}/\operatorname{Syl}_{p}(S_{m})|\cdot[H/H]$. Hence by assumption

[TABLE]

Therefore $\operatorname{char}^{H}_{K}\colon A(H)\to{\mathbb{Z}}$ sends both $[S_{m}]-|S_{m}|\cdot[H/H]$ and $[S_{m}/\operatorname{Syl}_{p}(S_{m})]-|S_{m}/\operatorname{Syl}_{p}(S_{m})|\cdot[H/H]$ to elements in $p{\mathbb{Z}}$. Since $\operatorname{char}_{K}^{H}([S_{m}]-|S_{m}|\cdot[H/H])=0-|S_{m}|$ for $K\not=\{1\}$, we conclude that $K=\{1\}$ or that $p\not=0$. If $K=\{1\}$, then ${\mathbb{I}}(H)=\mathcal{P}(\{1\},0)$ is contained in $\mathcal{P}(K,p)$. Suppose that $K\not=\{1\}$. Then $p$ is a prime. We have

[TABLE]

Since this integer must belong to $p{\mathbb{Z}}$ and $|S_{m}/\operatorname{Syl}_{p}(S_{m})|$ is prime to $p$, we get $\left(S_{m}/\operatorname{Syl}_{p}(S_{m})\right)^{K}\not=\emptyset$. Hence $K$ must be a $p$-group. This implies $\mathcal{P}(K,p)=\mathcal{P}(\{1\},p)$ and therefore ${\mathbb{I}}(H)=\mathcal{P}(\{1\},0)\subseteq\mathcal{P}(K,p)$. This finishes the proof of condition (3).

Condition (4) follows from the proof of the Segal Conjecture for a finite group $H$ due to Carlsson [6]. This finishes the proof of Theorem 0.2. ∎

6. An improved Strategy for a Proof of a Completion Theorem

The next result follows from Theorem 4.1, Lemma 4.4 and a construction of a modified Chern character analogous to the one in [11, Theorem 4.6 and Lemma 6.2] which will ensure that the condition about the differentials in the equivariant Atiyah-Hirzebruch spectral sequence appearing in Lemma 4.4 is satisfied. We do not give more details here, since the interesting case of the Segal Conjecture and of the Atiyah-Segal Completion Theorem are already covered by Theorem 0.2 and by [15].

Let $G$ be a (discrete) group. Let $\mathcal{F}$ be a family of subgroups of $G$ such that there is an upper bound on the orders of the subgroups appearing $\mathcal{F}$. Let $\mathcal{H}^{?}_{*}$ be an equivariant cohomology theory with values in $R$-modules which satisfies the disjoint union axiom. Define a contravariant functor

[TABLE]

with the category $\EuR{FGINJ}$ of finite groups with injective group homomorphism as source by sending an injective homomorphism $\alpha\colon H\to K$ to the composite

[TABLE]

where $\operatorname{pr}\colon H/K=\operatorname{ind}_{\alpha}(\ast)\to\ast$ is the projection and $\operatorname{ind}_{\alpha}$ comes from the induction structure of $\mathcal{H}_{?}^{*}$. Assume that $\mathcal{H}^{?}_{*}$ comes with a multiplicative structure.

Theorem 6.2** **(Improved Strategy for the proof of

Theorem 0.2).

Suppose that the following conditions are satisfied.

1. (1)

The ring $\mathcal{H}^{0}(\ast)$ is Noetherian; 2. (2)

Let $H\subseteq G$ be any finite subgroup and $m\in{\mathbb{Z}}$ be any integer. Then the $\mathcal{H}^{0}(\ast)$-module $\mathcal{H}^{m}_{H}(\ast)$ is finitely generated, there exists an integer $C(H,m)$ such that multiplication with $C(H,m)$ annihilates the torsion-submodule $\operatorname{tors}_{{\mathbb{Z}}}(\mathcal{H}^{m}_{H}(\ast))$ of the abelian group $\mathcal{H}^{m}_{H}(\ast)$ and the $R$-module $\mathcal{H}^{m}_{H}(\ast)/\operatorname{tors}_{{\mathbb{Z}}}(\mathcal{H}^{m}_{H}(\ast))$ is projective; 3. (3)

Let $H$ be any element of $\mathcal{F}$. Let $\mathcal{P}\subseteq\mathcal{H}^{0}_{H}(\ast)$ be any prime ideal. Then the augmentation ideal

[TABLE]

is contained in $\mathcal{P}$ if $\mathcal{P}$ contains the image of the structure map for $H$ of the inverse limit

[TABLE] 4. (4)

The Completion Theorems is true for every finite group $H$ in the case $X=L=\ast$ and $f=\operatorname{id}\colon\ast\to\ast$, i.e., for every finite group $H$ the map of pro-$\mathcal{H}^{0}(\ast)$-modules

[TABLE]

defined in (3.3) is an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules; 5. (5)

The covariant functor (6.1) extends to a Mackey functor.

Then the Completion Theorem is true for $\mathcal{H}^{?}_{*}$ and every $G$-map $f\colon X\to L$ from a finite proper $G$-$CW$-complex $X$ to a proper finite dimensional $G$-$CW$-complex $L$ with the property that there is an upper bound on the order of its isotropy groups. $L$, i.e., the map of pro-$\mathcal{H}^{0}(\ast)$-modules

[TABLE]

defined in (3.3) is an isomorphism of pro-$\mathcal{H}^{0}(\ast)$-modules.

Remark 6.3**.**

The advantage of Theorem 6.2 in comparison with Theorem 4.1 is that the conditions do not involve $L$ and $f\colon X\to L$ anymore and do only depend on the functor $\mathcal{H}^{q}_{?}(\ast)\colon\EuR{FGINJ}\to{\mathbb{Z}}\text{ -}\EuR{MODULES}$. If one considers the case $R={\mathbb{Z}}$ and assumes $\mathcal{H}^{0}(\ast)={\mathbb{Z}}$, then condition (1) is obviously satisfied and condition (2) reduces to the condition that for any finite subgroup $H\subseteq G$ and any integer $m\in{\mathbb{Z}}$ the abelian group $\mathcal{H}^{m}_{H}(\ast)$ is finitely generated.

Remark 6.4** (Family version).**

We mention without proof that there is a also a family version of Theorem 0.2. Its formulation is analogous to the one of the family version of the Atiyah-Segal Completion Theorem for infinite groups, see [14, Section 6].