Stabilization of the homotopy groups of the self equivalences of linear spheres

TL;DR

This paper proves that the homotopy groups of the space of $G$-equivariant self-maps of linear spheres stabilize and describes their stable structure in terms of classifying spaces of isotropy groups.

Contribution

It establishes the stabilization of homotopy groups for self-equivalence spaces of linear spheres under certain representation conditions and characterizes the stable groups explicitly.

Findings

Homotopy groups stabilize as the dimension increases.

Stable groups are described as direct sums over conjugacy classes of isotropy groups.

Provides a formula for the stable homotopy groups in terms of classifying spaces.

Abstract

Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}^n U_n$ and $S(V_n)$ denote the linear sphere of unit vectors. Then for any $i \geq 0$ the sequence of group $\dots \rightarrow \pi_i \operatorname{map}^G(S(V_n),S(V_n)) \rightarrow \pi_i \operatorname{map}^G(S(V_{n+1}),S(V_{n+1})) \rightarrow \dots$ stabilizes with the stable group $\oplus_H \omega_i(BW_GH)$ where $H$ runs through representatives of the conjugacy classes of all the isotropy group of the points of $S(\oplus_n U_n)$.

Equations

$${\mathrm{map}}^G(S(U_{\le n}),S(U_{\le n}))\stackrel{f\mapsto f\ast {\mathrm{id}}_{S(U_{n+1})}}{\to }{\mathrm{map}}^G(S(U_{\le n+1}),S(U_{\le n+1}))$$

$${\pi }_i{\mathrm{map}}^G(S(U_{\le n}),S(U_{\le n}))\cong \underset{(H)\subseteq \mathcal{F}(U_{\bullet })}{⨁}{\omega }_i(BWH)$$

$${\mathrm{map}}^G(S(U),S(U))\stackrel{\mathrm{susp}:f\mapsto \Sigma f}{\to }{\mathrm{map}}^G(S^U,S^U;{\mathrm{id}}_{\{{\alpha }_0,{\alpha }_1\}}).$$

$${\mathrm{map}}_{\ast }^G(S^{U_{\le n}},S^{U_{\le n}})\stackrel{f\mapsto f\wedge S^{U_{n+1}}}{\to }{\mathrm{map}}_{\ast }^G(S^{U_{\le n+1}},S^{U_{\le n+1}})$$

$${\mathrm{colim}}_n{\pi }_i{\mathrm{map}}_{\ast }^G(S^{U_{\le n}},S^{U_{\le n}})\cong \underset{(H)\subseteq \mathcal{F}(U_{\bullet })}{\oplus }{\omega }_i(BWH).$$

$${\mathrm{colim}}_n{\pi }_i{\mathrm{map}}_{\ast }^G(S^{U_{\le n}},S^{U_{\le n}})\cong \underset{(H)\subseteq \mathcal{F}(U_{\bullet })}{\oplus }{\omega }_i(BWH\coprod BWH).$$

$$\mathrm{conn}X=k.$$

$$\mathrm{Fib}(p_1,b_1)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E_1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$F_1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Iso}}_G(X,Y)=\{G_x:x\in X\setminus Y\}.$$

$${\dim }_H(A,B)\stackrel{\text{def}}{=}\dim (A^H,B^H).$$

$$\mathrm{Fib}(i^{\ast },\phi )\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{map}}^G(A,X)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\coprod }_{i\in \mathcal{A}}{S}^{n-1}\times G/H_i\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{map}}^G(A,T)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\prod }_i\mathrm{map}(D^n,X^{H_i})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\alpha }_0,{\alpha }_1\in \Sigma X,$$

$$F(\Sigma X,Y;y_0,y_1)\cong F(X,{\mathbb{P}}_{y_0,y_1}Y).$$

$${\eta }_X:X\to {\mathbb{P}}_{{\alpha }_0,{\alpha }_1}\Sigma X({\eta }_X(x)(t)=(t,x)\in \Sigma X)$$

$${\mathrm{map}}^G(\Sigma X,\Sigma X)\cong {\mathrm{map}}^G(X,{\mathbb{P}}_{{\alpha }_0,{\alpha }_1}\Sigma X).$$

$$F(S^n,S^m)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{A}=\qquad\vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 13.70012pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-8.15001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 13.29996pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{a_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.15001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-13.70012pt\raise-19.31888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{a_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.15001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 40.30002pt\raise-19.31888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{a_{20}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.30002pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-8.15001pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 13.29996pt\raise-43.78914pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{a_{10}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.15001pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.15001pt\raise-38.63776pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A_{0}}$}}}}}}}\ignorespaces}}}}\ignorespaces}$$

$$(\mathcal{A},\underline{x})\qquad=\qquad\vbox{ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 13.70012pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\&&\\&&\crcr}}}\ignorespaces{\hbox{\kern-8.15001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 13.29996pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{a_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.15001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-13.70012pt\raise-19.31888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8625pt\hbox{$\scriptstyle{a_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-28.80444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.15001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 40.30002pt\raise-19.31888pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 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$$\Lambda (\mathcal{A},\underline{x})\stackrel{\text{def}}{=}\mathrm{holim}(\mathcal{A},\underline{x}).$$

$$\Lambda (\mathcal{A},\underline{x})\cong \mathrm{hoFib}\left(\mathrm{hoFib}(a_{32},x_2)\stackrel{(a_{31},a_{20})}{\to }\mathrm{hoFib}(a_{10},x_0),x_1\right)$$

$$A_3\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{hoFib}(a_{31},x_1)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Lambda (\mathcal{A},\underline{x})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\coprod _{\varnothing \ne \sigma \subseteq [n]}{\Delta }^{|\sigma |-1}\times X_{\sigma }$$

Full text

1. Introduction

Let $G$ be a finite group. A real representation $U$ of $G$ can be equipped with an essentially unique $G$-invariant norm. The set $S(U)$ of unit vectors is called a linear sphere for $G$. The one point compactification of $U$ is denoted $S^{U}$ with $\infty\in S^{U}$ as a basepoint.

This paper grew out of the interest in stabilization properties of the homotopy groups of the space $\operatorname{map}^{G}(S(U),S(U))$ of equivariant self maps. To make this precise, let $U_{1},U_{2},\dots$ be a sequence of real representations of $G$. Let $\operatorname{Irr}(U_{\bullet})$ be the set of their irreducible summands. Throughout we will assume:

1. (U)

Any $V\in\operatorname{Irr}(U_{\bullet})$ has infinite multiplicity in $\oplus_{n\geq 1}U_{n}$.

A map of unpointed spaces $f\colon X\to Y$ is called a $k$-equivalence if it induces a bijection on path components, isomorphisms $\pi_{i}X\to\pi_{i}Y$ for all $1\leq i\leq k$ and an epimorphism $\pi_{k+1}X\to\pi_{k+1}Y$ for any choice of basepoint in $X$. Let $\omega_{i}(X)$ denote the stable homotopy groups of $X_{+}$ (the disjoint union of $X$ with a basepoint). Let $BG$ denote the classifying space of a group $G$. Let $(H)$ denote the conjugacy class of $H\leq G$ and $WH=N_{G}(H)/H$. Let $\operatorname{Iso}_{G}(X)$ be the set of the isotropy groups of the points of a $G$-space $X$.

If $U,V$ are orthogonal representations of $G$ then $S(U\oplus V)$ is homeomorphic to the join $S(U)*S(V)$. There results $\operatorname{map}^{G}(S(U),S(U))\xrightarrow{f\mapsto f*\operatorname{id}_{S(V)}}\operatorname{map}^{G}(S(U\oplus V),S(U\oplus V))$.

Theorem 1.1**.**

Let $U_{1},U_{2},\dots$ be a sequence of real representations of a finite group $G$ which satisfies hypothesis (U). Let $U_{\leq n}$ denote $\oplus_{i=1}^{n}U_{i}$. Then for any $k\geq 0$ the maps

[TABLE]

are $k$-equivalences for all sufficiently large $n$. In addition, provided $n$ is sufficiently large, there are isomorphisms (bijection if $i=0$) for any $0\leq i\leq k$ and any choice of basepoint

[TABLE]

where $\mathcal{F}(U_{\bullet})$ is the smallest collection of subgroups of $G$ which contains $\cup_{V\in\operatorname{Irr}(U_{\bullet})}\operatorname{Iso}_{G}(S(V))$ and is closed under intersection of groups, and the sum is over its conjugacy classes.

Unreduced suspension gives a homeomorphism $\Sigma S(U)\cong S^{U}$. Denote the fixed points $0,\infty\in S^{U}$ by $\alpha_{0},\alpha_{1}$, the latter is the basepoint of $S^{U}$. Let $\operatorname{map}^{G}(S^{U},S^{U};\operatorname{id}_{\{\alpha_{0},\alpha_{1}\}})$ be the space of self maps which leave $\alpha_{0}$ and $\alpha_{1}$ fixed.

Proposition 1.2**.**

With the set up and notation of Theorem 1.1, let $U=U_{\leq n}$ for sufficiently large $n$. Then unreduced suspension gives a $k$-equivalence

[TABLE]

If $U,V$ are representations, there is a homeomorphism $S^{U}\wedge S^{V}\xrightarrow{u\wedge v\mapsto u+v}S^{U+V}$.

Theorem 1.3**.**

With the set up and notation of Theorem 1.1, for every sufficiently large $n$, the map (between the pointed mapping spaces)

[TABLE]

is a $k$-equivalence. If $Irr(U_{\bullet})$ contains the trivial representation then for all $i\geq 0$ there are isomorphisms (bijection if $i=0$)

[TABLE]

If $Irr(U_{\bullet})$ does not contain the trivial representation then

[TABLE]

1.4**.**

Relation with tom Dieck splitting. Let $\mathbf{S}$ be the sphere spectrum in a category of $G$-spectra on a complete universe [6, Sec. I.2]. tom Dieck’s splitting [12, Satz 2] yields $\pi_{*}(\mathbf{S}^{G})\cong\oplus_{H}\,\omega_{*}(BWH)$ where $H$ runs through representatives of the conjugacy classes of all the subgroups of $G$. Theorem 1.3 can be viewed as an extension of this result to non-complete $G$-universes, and in fact, “universes” which do not contain the trivial representation.

Indeed, hypothesis (U) makes $U_{\leq n}$ an indexing sequence in a $G$-universe $\mathcal{U}$ underlying a category of $G$-spectra in the sense of [6, Sec. I.2], with the only except that we do not insist that $\operatorname{Irr}(U_{\bullet})$ contains the trivial representation. If $\operatorname{Irr}(U_{\bullet})$ contains all the irreducible representations of $G$ then $\mathcal{U}$ is a complete $G$-universe and Theorem 1.3 recovers tom Dieck’s splitting since $\operatorname{Irr}(U_{\bullet})$ contains all $\operatorname{Irr}(\operatorname{Ind}_{H}^{G}(\mathbb{R}))$, since $\mathbf{S}\simeq F(\mathbf{S},\mathbf{S})$, and since sequential colimits commute with $G$-fixed points and homotopy groups.

If the trivial representation is present in $\operatorname{Irr}(U_{\bullet})$ then $S(U_{\leq n})$ has a fixed point and Theorem 1.1 can be deduced from Hauschild’s results [3, Satz 2.4]. Things are harder in its absence. Becker and Schultz proved the theorem in the case that $G$ acts freely [1] using geometric methods. Spectral sequence arguments were used by Schultz [9, Prop. 6.5] to prove Theorem 1.1 when $G$ is cyclic. Klaus proved that for any $k\geq 1$ the groups $\pi_{k}\operatorname{map}^{G}(S(U_{\leq n}),S(U_{\leq n}))$, where $\operatorname{id}$ is the basepoint, are finite for all sufficiently large $n$ [5, Proposition 2.5]. The author improved this result in [7], giving bounds for their order (uniform in $n$ for each $k$). However, to establish Theorem 1.1 in its full generality spectral sequences become unmanageable.

The stabilization statement in Theorem 1.1 is a consequence of Proposition 6.1 while the identification of the limit groups is a special case of Proposition 7.3. These propositions are the main technical results of this paper. They allow, in principle, to prove Theorem 1.1 for a larger class of spaces than linear spheres. Their hypotheses, though, are very restrictive.

Acknowledgements: I would like to thank Michael Crabb who read early versions of this paper and shared his ideas and most importantly, suggesting the description of the limit groups in Theorem 1.1 as a direct sum of stable homotopy groups. I would also like to thank Irakli Patchkoria for helpful discussions.

2. Preliminaries

Definition 2.1**.**

A map $f\colon X\to Y$ of spaces is called a $k$-equivalence if it is bijective on components and for any $x\in X$ the induced maps $\pi_{i}(X,x)\to\pi_{i}(Y,f(x))$ are isomorphisms for all $1\leq i\leq k$ and epimorphism for $k+1$.

This is just a (convenient) “shift by $1$” of the standard definitions of $n$-connectedness of maps, see [14]. A space $X$ is called $k$-connected, where $k\geq 0$, if $X\to*$ is a $k$-equivalence. We will write

[TABLE]

By convention $\operatorname{conn}X=-1$ if the number of path components of $X$ is not $1$. The next two results are straightforward.

Lemma 2.2**.**

Let $k\geq 0$. Consider a morphism of fibre sequences where $b_{1}\in B_{1}$.

[TABLE]

1. (1)

If $f$ is a $k$-equivalence and the map $h$ is a $k$-equivalence for any choice of $b_{1}\in B_{1}$ then $g$ is a $k$-equivalence. 2. (2)

If $f$ is a $(k+1)$-equivalence and $g$ is a $k$-equivalence then $h$ is a $k$-equivalence for any choice of $b_{1}\in B_{1}$.

Proof.

This is standard diagram chase of exact sequences of pointed sets and groups. The first assertion is slightly more delicate in connection to the surjection on components and uses the homotopy lifting property of $p_{2}$. ∎

Lemma 2.3**.**

Let $k\geq 0$ and consider the following ladder of Serre fibrations

[TABLE]

Assume that $f,g,h$ induce $k$-equivalences on the fibres of $p_{1}$ and $p_{2}$ and on the fibres of $q_{1}$ and $q_{2}$. Then they induce $k$-equivalences on the fibres of $p_{1}\circ q_{1}$ and $p_{2}\circ q_{2}$.

Proof.

Choose $b_{1}\in B_{1}$ and set $b_{2}=f(b_{2})$. We need to show that $F_{i}=\operatorname{Fib}(p_{i}\circ q_{i},b_{i})$ are $k$-equivalent. Set $X_{i}=\operatorname{Fib}(p_{i},b_{i})$. We obtain a morphism of fibrations

[TABLE]

By hypothesis $g|_{X_{1}}$ is a $k$-equivalence, so by Lemma 2.2(1) it remain to show that all the fibres of the rows of this diagram are $k$-equivalent. These fibres are equal to the fibres of $q_{1}$ and $q_{2}$ and by the hypothesis they are $k$-equivalent. ∎

Throughout this paper we will work in a “convenient category of $G$-spaces”, that is the category $\mathcal{CGWH}$ of compactly generated weak Hausdorff spaces, or the category $\mathcal{CGH}$ of compactly generated Hausdorff spaces, see [10] or [11]. This category has products and function complexes $F(X,Y)$ giving adjunction homeomorphisms $\operatorname{map}(Z\times X,Y)\cong\operatorname{map}(Z,F(X,Y))$ where $\operatorname{map}$ denotes the set of morphisms in $\mathcal{CGWH}$. In fact, $\mathcal{CGWH}$ is enriched over itself and $\operatorname{map}(X,Y)\cong F(X,Y)$.

Let $G$ be a discrete group, e.g finite. Let $G-\mathcal{CGWH}$ be the category of $G$-spaces. Regarding $X$ and $Y$ as objects in $\mathcal{CGWH}$ via the forgetful functor, $F(X,Y)$ is equipped with a standard action of $G$ where $(g\cdot\varphi)(x)=g\varphi(g^{-1}x)$. In this way the set $\operatorname{map}^{G}(X,Y)$ of all $G$-maps $X\to Y$ is equipped with a topology giving rise to the adjunction homeomorphism $\operatorname{map}^{G}(Z\times X,Y)\cong\operatorname{map}^{G}(Z,F(X,Y))$.

Let $Y\subseteq X$ be an inclusion of $G$-spaces. We denote by $G_{x}$ the stabilizer of $x\in X$. Set

[TABLE]

If $Y=\emptyset$ we will simply write $\operatorname{Iso}_{G}(X)$.

An inclusion $B\subseteq A$ of $G$-spaces is a relative $G$-CW complex if $A$ is obtained from $B$ by attaching equivariant cells. A $G$-CW complex is a space obtained in this way from the empty set. We emphasize that by $G$-CW complexes we always mean that $G$ acts cellularly (by permuting cells). See [13, Chapter II.1]. For $H\in\operatorname{Iso}_{G}(A,B)$, the $H$-relative dimension is

[TABLE]

This is the maximum dimension of an equivariant cell of type $G/H$ in $A$ which is not contained in $B$. For any $G$-space $Y$, a relative $G$-CW complex $(A,B)$ gives rise to a Serre fibration $\operatorname{map}^{G}(A,Y)\to\operatorname{map}^{G}(B,Y)$.

A map of $G$-spaces is a $k$-equivalence if this is the case by forgetting the action of $G$.

Lemma 2.4**.**

Fix some $k\geq 0$. Suppose $B\subseteq A$ is an inclusion of finite dimensional $G$-CW complexes and $f\colon X\to Y$ is a map of $G$-spaces. For any $H\in\operatorname{Iso}_{G}(A,B)$ set $n_{H}=\dim_{H}(A,B)$ and assume that the map $X^{H}\xrightarrow{f^{H}}Y^{H}$ is a $(k+n_{H})$-equivalence. Then for any $\varphi\in\operatorname{map}^{G}(B,X)$ the map $f_{*}$ induced on fibres in

[TABLE]

is a $k$-equivalence of spaces.

Proof.

We use induction on $n=\dim(A,B)$. If $n=-1$ then $A=B$ and the result is trivial. Assume that $n\geq 0$ and let $A^{\prime}\subseteq A$ be the union of the $(n-1)^{\text{st}}$ skeleton of $A$ with $B$. Let $i\colon B\to A$ and $j\colon B\to A^{\prime}$ and $\ell\colon A^{\prime}\to A$ denote the inclusions. We obtain a diagram of fibrations

[TABLE]

such that composition of the rows are the maps $i^{*}$. By construction $\dim(A^{\prime},B)\leq n-1$ and also $\dim_{H}(A^{\prime},B)\leq\dim_{H}(A,B)=n_{H}$ for any $H\in\operatorname{Iso}_{G}(A^{\prime},B)$. The induction hypothesis applies to the inclusion $B\subseteq A^{\prime}$ and we deduce that the fibres of $j^{*}$ in the second square are $k$-equivalent. By Lemma 2.3 it remains to show that the fibres of the maps $\ell^{*}$ are $k$-equivalent.

Since $A$ is obtained from $A^{\prime}$ by attaching equivariant $n$-cells, we get a pushout diagram

[TABLE]

where $\mathcal{A}$ indexes the equivariant $n$-cells attached to $A^{\prime}$, hence $n_{H_{i}}=n$ for all $i\in\mathcal{A}$. By applying $\operatorname{map}^{G}(-,T)$, where $T$ is any $G$-space, we obtain a pullback diagram, natural in $T$,

[TABLE]

Since this is is a pullback square, the fibres of the horizontal maps are homeomorphic. Applying this for $T=X$ and $T=Y$, it remains to prove that the maps on all fibres in the following commutative diagram

[TABLE]

are $k$-equivalences. If $n=0$ then the spaces on the right are points and the map of the spaces on the left is by hypothesis $n_{H_{i}}+k=n+k=k$ equivalence, so the map on fibres is a $k$-equivalence. If $n\geq 1$ then by the hypothesis the vertical arrow on the left is a $(k+n)$-equivalence, hence a $k$-equivalence, and the vertical arrow on the right is a $k+n-(n-1)=k+1$ equivalence. Lemma 2.2 shows that the map on all fibres is a $k$-equivalence and this completes the proof. ∎

Corollary 2.5**.**

Let $k\geq 0$. Let $A$ be a finite $G$-CW complex and $B$ be a subcomplex. Let $Y$ be a $G$ space and let $k\geq 0$. Assume that $\operatorname{conn}Y^{H}-\dim_{H}(A,B)\geq k$ for every $H\in\operatorname{Iso}_{G}(A,B)$. Then $\operatorname{map}^{G}(A,Y)\xrightarrow{i^{*}}\operatorname{map}^{G}(B,Y)$ is a $k$-equivalence.

Proof.

Lemma 2.4 with $B\subseteq A$ and $Y\to*$ shows that all the fibres of $\operatorname{map}^{G}(A,Y)\to\operatorname{map}^{G}(B,Y)$ are $k$-connected. Then apply Lemma 2.2(1) with $g=i^{*}$ and $f$ the identity on $\operatorname{map}^{G}(B,Y)$. ∎

2.6**.**

Let $\Sigma X$ be the unreduced suspension of a space $X$. The images of $0,1\in I$ yield two distinguished points

[TABLE]

of which $\alpha_{1}$ is chosen as the basepoint. If $X$ is a $G$-space both $\alpha_{0},\alpha_{1}$ are fixed points.

Let $Y$ be a space and $y_{0},y_{1}\in Y$ points. Let $F(\Sigma X,Y;y_{0},y_{1})$ be the subspace of maps $f\colon\Sigma X\to Y$ such that $f(\alpha_{0})=y_{0}$ and $f(\alpha_{1})=y_{1}$. Let $\mathbb{P}_{y_{0},y_{1}}Y$ denote the space of paths $\omega\colon I\to Y$ such that $\omega(0)=y_{0}$ and $\omega(1)=y_{1}$. There is an adjunction homeomorphism

[TABLE]

In particular $F(\Sigma X,\Sigma X;\alpha_{0},\alpha_{1})\cong F(X,\mathbb{P}_{\alpha_{0},\alpha_{1}}\Sigma X)$ and denote the adjoint of the identity by

[TABLE]

If $X$ is a $G$-space then $\eta_{X}$ is a $G$ map, $(\eta_{X})^{H}=\eta_{X^{H}}$ for any $H\leq G$, and $\eta_{X}$ the unit of the adjunction isomorphism

[TABLE]

We will need the following (somewhat expected) corollary of Freudenthal’s theorem.

Proposition 2.7**.**

1. (a)

The map $\eta_{S^{n}}\colon S^{n}\to\mathbb{P}_{\alpha_{0},\alpha_{1}}\Sigma S^{n}$ is a $(2n-2)$-equivalence, where $n\geq 1$. 2. (b)

Suppose $m>n\geq 1$. Then $F(S^{n},S^{m})\xrightarrow{\operatorname{susp}\colon f\mapsto\Sigma f}F(\Sigma S^{n},\Sigma S^{m})$ is a $(m-1)$-equivalence of path connected spaces.

Proof.

(a) Choose a basepoint $x_{0}\in S^{n}$ and consider the quotient map $\pi\colon\Sigma S^{n}\to\Sigma^{\operatorname{red}}S^{n}$. By inspection $\pi\circ\eta_{S^{n}}$ is the canonical map $S^{n}\to\Omega\Sigma^{\operatorname{red}}S^{n}$ which is a $(2n-2)$-equivalence by Freudenthal’s theorem. The result follows since $\pi$ is a homotopy equivalence.

(b) By definition of $\eta_{S^{n}}$ the triangle in the following diagram commutes

[TABLE]

The inclusion of the fibre in the second row is a $(m-1)$-equivalence. The vertical arrow is a $(2m-2-n)$-equivalence by part (a) and Lemma 2.4 applied to $\emptyset\subseteq S^{n}$ and $\eta_{S^{m}}$, and $F(S^{n},S^{m})$ is clearly path connected. Since $m-1\leq 2m-2-n$, the result follows. ∎

3. Square diagrams of spaces

Let $\mathbb{P}X=F(I,X)$ denote the path space of $X$. The homotopy pullback of a diagram of spaces $X_{0}\xrightarrow{f}X_{2}\xleftarrow{g}X_{1}$ is the subspace of $X_{0}\times X_{1}\times\mathbb{P}X_{2}$ consisting of $(x_{0},x_{1},\omega)$ such that $f(x_{0})=\omega(0)$ and $g(x_{1})=\omega(1)$. The homotopy fibre of $X\xrightarrow{f}Y$ over $y_{0}\in Y$ is the homotopy pullback of $X\xrightarrow{f}Y\xleftarrow{y_{0}}*$. There is an inclusion $\operatorname{Fib}(f,y_{0})\subseteq\operatorname{hoFib}(f,y_{0})$ via the constant paths, and if $f$ is a Serre fibration this inclusion is a weak homotopy equivalence.

Definition 3.1**.**

Let $\mathfrak{S}$ be the category whose objects are commutative diagrams of spaces

[TABLE]

Morphisms are natural transformations of diagrams. Thus, a morphism $\varphi\colon\mathcal{A}\to\mathcal{B}$ is a quadruple $(\varphi_{0},\varphi_{1},\varphi_{2},\varphi_{3})$ of maps $\varphi_{i}\colon A_{i}\to B_{i}$ with the obvious commutation relations with the structure maps of $\mathcal{A}$ and $\mathcal{B}$.

A basepoint for $\mathcal{A}$ is a triple of $\underline{x}=(x_{0},x_{1},x_{2})\in A_{0}\times A_{1}\times A_{2}$ such that $a_{20}(x_{2})=x_{0}$ and $a_{10}(x_{1})=x_{0}$. Notice that we do not choose $x_{3}\in A_{3}$ compatible with $x_{0},x_{1},x_{2}$.

A basepoint of $\mathcal{A}\in\mathfrak{S}$ gives rise to the following diagram of spaces which we denote by $(\mathcal{A},\underline{x})$.

[TABLE]

We obtain a category $\mathfrak{S}_{*}$ whose objects are $(\mathcal{A},\underline{x})$ with natural transformations between them. The homotopy limit functor gives rise to a functor $\Lambda\colon\mathfrak{S}_{*}\to\textbf{Spaces}$

[TABLE]

Fubini’s theorem for homotopy limits [2, Secs. 24 and 31] implies that $(\mathcal{A},\underline{x})$ can be calculated by first taking the homotopy limits of the rows (resp. columns) and then take the homotopy limits of the resulting pullback diagram of spaces. Therefore

[TABLE]

where $x_{1}\in\operatorname{Fib}(a_{10},x_{0})\subseteq\operatorname{hoFib}(a_{10},x_{0})$ and $x_{2}\in\operatorname{Fib}(a_{20},x_{0})\subseteq\operatorname{hoFib}(a_{20},x_{0})$.

Lemma 3.2**.**

Let $\varphi\colon\mathcal{A}\to\mathcal{B}$ be a morphism in $\mathfrak{S}$ depicted by the vertical arrows in

[TABLE]

1. (1)

If the side faces (or the back and front faces) are homotopy pullback squares then the induced map $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\varphi)}\Lambda(\mathcal{B},\varphi(\underline{x}))$ is a (weak) homotopy equivalence for any choice of basepoint $\underline{x}$ for $\mathcal{A}$. 2. (2)

Let $\underline{x}$ be a basepoint for $\mathcal{A}$. Suppose that

• (i)

$\varphi_{2}$* and $\varphi_{0}$ induce a $(k+1)$-equivalence $\operatorname{hoFib}(a_{20},x_{0})\to\operatorname{hoFib}(b_{20},\varphi_{0}(x_{0}))$ and*

• (ii)

$\varphi_{1},\varphi_{3}$* induce a $k$-equivalence $\operatorname{hoFib}(a_{31},x_{1})\to\operatorname{hoFib}(b_{31},\varphi_{1}(x_{1}))$.*

Then $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\varphi)}\Lambda(\mathcal{B},\varphi(\underline{x}))$ is a $k$-equivalence. 3. (3)

Suppose that $A_{2}\xrightarrow{a_{20}}A_{0}$ is a Serre fibration and that

• (i)

$\operatorname{hoFib}(a_{20},x_{0})\to\operatorname{hoFib}(b_{20},\varphi_{1}(x_{0}))$* is a $k$-equivalence for any basepoint $x_{0}\in A_{0}$, and*

• (ii)

$\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\varphi)}\Lambda(\mathcal{B},\varphi(\underline{x}))$* is a $k$-equivalence for any choice of basepoint $\underline{x}$ in $\mathcal{A}$.*

Then $\operatorname{hoFib}(a_{31},x_{1})\to\operatorname{hoFib}(b_{31},\varphi(x_{1}))$ is a $k$-equivalence for any basepoint $x_{1}\in A_{1}$.

Proof.

(1) Since each side face is a homotopy pullback square, the induced maps on homotopy fibres of its rows is a weak equivalence. Thus, the vertical arrows in

[TABLE]

are weak homotopy equivalences. Therefore the map induced on the homotopy fibres of the rows over $x_{2}$ and $\varphi_{2}(x_{2})$ are weak equivalences, and the result follows from (3).

(2) This is immediate from (3) and Lemma 2.2(2).

(3) Choose $x_{1}\in A_{1}$ and set $x_{0}=a_{10}(x_{1})$. There results a commutative diagram as in (3.4). By the hypothesis the vertical arrow on the right of (3.4) is a $k$-equivalence, and our goal is to show the the same is true for the vertical arrow on the left. By Lemma 2.2(1) it remain to show that for any $x_{2}\in\operatorname{hoFib}(a_{20},x_{0})$ the map induced on the homotopy fibres of the horizontal arrows is a $k$-equivalence. By hypothesis $A_{2}\to A_{0}$ is a Serre fibration, so the inclusion $\operatorname{fib}(a_{20},x_{0})\subseteq\operatorname{hoFib}(a_{20},x_{0})$ is a weak homotopy equivalence. Therefore we may consider only $x_{2}\in\operatorname{fib}(a_{20},x_{0})$ in which case $\underline{x}=(x_{0},x_{1},x_{2})$ forms a basepoint for $\mathcal{A}$ and it follows from (3) that the map of the homotopy fibres over $x_{2}$ and $\varphi_{2}(x_{2})$ in the diagram (3.4) is the map $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\varphi)}\Lambda(\mathcal{B},\varphi(\underline{x}))$ which by hypothesis is a $k$-equivalence. This completes the proof. ∎

Given an object $\mathcal{A}\in\mathfrak{S}$ let $I\times\mathcal{A}$ be the object in $\mathfrak{S}$ obtained by applying the functor $I\times-$ objectwise. Similarly $\mathbb{P}\mathcal{A}$ is obtained by applying the path space functor $\mathbb{P}(-)$ objectwise.

Definition 3.3**.**

Let $\mathcal{A},\mathcal{B}$ be objects in $\mathfrak{S}$. A homotopy is a morphism $\varphi\colon I\times\mathcal{A}\to\mathcal{B}$. We frequently refer to a homotopy as a family of morphisms $\varphi_{p}\colon\mathcal{A}\to\mathcal{B}$ (parameterized by $0\leq p\leq 1$).

The adjoint of a homotopy $\varphi\colon I\times\mathcal{A}\to\mathcal{B}$ is a morphism $\varphi^{\#}\colon\mathcal{A}\to\mathbb{P}\mathcal{B}$. If $\underline{x}$ is a basepoint of $\mathcal{A}$ then $\varphi^{\#}(\underline{x})$ is a basepoint in $\mathbb{P}\mathcal{B}$. Evaluation at $p\in I$ gives a morphism $(\mathbb{P}\mathcal{B},\varphi^{\#}(\underline{x}))\xrightarrow{\operatorname{ev}_{p}}(\mathcal{B},\varphi_{p}(\underline{x}))$ in $\mathfrak{S}_{*}$ which is an object-wise homotopy equivalence. We obtain a weak homotopy equivalence $\Lambda(\mathbb{P}\mathcal{B},\varphi^{\#}(\underline{x}))\xrightarrow{\Lambda(\operatorname{ev}_{p})}\Lambda(\mathcal{B},\varphi_{p}(\underline{x}))$. The following lemma is an immediate consequence.

Lemma 3.4**.**

Let $\varphi_{p}\colon\mathcal{A}\to B$ be a homotopy in $\mathfrak{S}$. Then for any basepoint $\underline{x}$ in $\mathcal{A}$ there is a commutative diagram in which both evaluation morphisms are (weak) homotopy equivalences

[TABLE]

In particular, if $\alpha,\beta\colon\mathcal{A}\to\mathcal{B}$ are homotopic morphisms and $\underline{x}$ of $\mathcal{A}$ is a basepoint then $\Lambda(\alpha,\underline{x})$ is a $k$-equivalence if and only if $\Lambda(\beta,\underline{x})$ is.

4. Join of spaces

The join $X_{1}*\dots*X_{n}$ is the homotopy colimit of the diagram of spaces indexed by the opposite category of the poset of the non-empty subsets $\sigma$ of $[n]=\{1,\dots,n\}$, and consisting of the spaces $X_{\sigma}\overset{\text{def}}{=}\prod_{i\in\sigma}X_{i}$ and projection maps between them. Let $\Delta^{n-1}=\{(t_{1},\dots,t_{n}):t_{i}\geq 0,\sum_{i}t_{i}=1\}$ be the standard $(n-1)$-simplex in $\mathbb{R}^{n}$. The underlying set of the join is the set of equivalence classes of

[TABLE]

where for any $\tau\subseteq\sigma$ we declare $(s_{i},x_{i})_{i\in\tau}\sim(t_{i},y_{i})_{i\in\sigma}$ if $x_{i}=y_{i}$ and $s_{i}=t_{i}$ for all $i\in\tau$ and $t_{i}=0$ for all $i\in\sigma\setminus\tau$. There are two natural choices to topologize the join, but when $X_{1},\dots,X_{n}$ are compact Hausdorff both agree with the quotient topology, see [8, Section 2].

4.1**.**

NOTATION: Since the join will play a key role in this paper we will write

[TABLE]

Its points are equivalence classes $[t_{1}x_{1},\dots,t_{n}x_{n}]$ where $(t_{1},\dots,t_{n})\in\Delta^{n-1}$ and it is understood that $t_{i}x_{i}$ may be omitted from the notation if either $X_{i}$ is empty or if $t_{i}=0$, and two such brackets represent the same point if they agree except in the entries where $t_{i}=0$.

Identify $\Delta^{1}$ with the unit interval $I$ via $I\xrightarrow{t\mapsto(t,1-t)}\Delta^{1}$. Then the join $XY$ of compact Hausdorff spaces fits in a pushout diagram

[TABLE]

If $X_{1},\dots,X_{n}$ are $G$-spaces then their join is also a $G$-space via the diagonal action. Given a $G$-space $Z$ we obtain a functor $X\mapsto XZ$ from $G-\mathcal{CGWH}$ to itself. This functor is, in fact, continuous in the sense that for any $G$-spaces $X,Y$ the resulting natural map

[TABLE]

is a continuous map. We can describe it explicitly: for any $f\in F(X,Y)$

[TABLE]

One easily checks that $J_{Z}$ is $G$-equivariant and passage to fixed points gives

[TABLE]

If $X,Y,Z$ are compact there are well known natural “associativity” homeomorphisms

[TABLE]

This allows us to identify, for example, $\operatorname{map}^{G}(AY,XY)\xrightarrow{J_{Z}}\operatorname{map}^{G}((AY)Z,(XY)Z)$ with $\operatorname{map}^{G}(AY,XY)\xrightarrow{J_{Z}}\operatorname{map}^{G}(AYZ,XYZ)$.

By inspection, these homeomorphisms together with (4.4) imply the commutativity of the following diagrams for compact $G$-spaces $A,T,Y,Z$.

[TABLE]

where $\operatorname{incl}\colon T\xrightarrow{t\mapsto[1\cdot t,0z]}TZ$ and $i\colon A\xrightarrow{a\mapsto[1\cdot a,0z]}AZ$ are the inclusions and we used the homeomorphism $(AY)Z\cong AYZ$.

Definition 4.2**.**

Let $A,X,Y,Z$ be compact $G$-spaces. Let

[TABLE]

be the unique map which renders the following diagram commutative

[TABLE]

It is clear that $\psi$ is natural in $A$. By inspection

[TABLE]

The remainder of this section is devoted to the definition and study of two maps

[TABLE]

of which the second is simply the quotient onto $A(YZ)\cong AYZ$. They arise in the computations in Section 6 in the context of the homeomorphisms (4.5), and the next definition is our starting point. Recall that $\Delta^{2}$ denotes the standard $2$-simplex in $\mathbb{R}^{3}$. For $i=0,1,2$ let $\partial_{i}\Delta^{2}$ denote the $i$th face of $\Delta^{2}$, i.e the elements $(t_{0},t_{1},t_{2})\in\Delta^{2}$ with $t_{i}=0$.

Definition 4.3**.**

Let $\tilde{\alpha},\tilde{\beta}\colon I\times I\to\Delta^{2}$ be the functions

[TABLE]

Both maps are clearly surjective, so for any $0\leq s,t,\leq 1$ there exist $0\leq s^{\prime},t^{\prime}\leq 1$ such that $\tilde{\beta}(s,t)=\tilde{\alpha}(s^{\prime},t^{\prime})$.

Proposition 4.4**.**

Define functions $s^{\prime},t^{\prime}\colon I\times I\to I$ as follows.

[TABLE]

1. (a)

$s^{\prime}$* is a continuous function, and $t^{\prime}$ is continuous away from $(0,0)$.* 2. (b)

$0\leq s^{\prime},t^{\prime}\leq 1$** 3. (c)

$\tilde{\beta}=\tilde{\alpha}\circ(s^{\prime},t^{\prime})$. 4. (d)

$s^{\prime}(0,t)=t$* and $t^{\prime}(s,0)=0$ and $t^{\prime}(s,1)=1$. Also $t^{\prime}(0,t)=1$ for all $t\neq 0$.*

Proof.

First, $s+t-st=1-(1-s)(1-t)$. This shows that $0\leq s^{\prime}\leq 1$ and that the denominator in the formula for $t^{\prime}$ vanishes if and only if $s=t=0$. This shows that $t^{\prime}$ is well defined and that it is continuous away from $(0,0)$. The continuity of $s^{\prime}$ is clear Also, $s+t-st=t+s(1-t)\geq t$ which shows that $0\leq t^{\prime}\leq 1$. This proves items (a) and (b). Items (c) and (d) follow by inspection of the formulas. ∎

Of course, the maps $\tilde{\alpha}$ and $\tilde{\beta}$ are homotopic for trivial reasons. But we will need an explicit homotopy satisfying some conditions. Given a homotopy $h\colon I\times X\to Y$ we will write $h_{p}\colon X\to Y$ for the restriction of $h$ to $\{p\}\times X$ where $0\leq p\leq 1$.

Proposition 4.5**.**

There exists a homotopy $\tilde{\theta}\colon I^{2}\times I\to\Delta^{2}$ from $\tilde{\alpha}$ to $\tilde{\beta}$, written as a family of maps $\tilde{\theta}_{p}\colon I^{2}\to\Delta^{2}$ parameterized by $0\leq p\leq 1$, with the properties

[TABLE]

Proof.

Define functions $S,T\colon I\times I^{2}\to I$ by

[TABLE]

It is clear from Proposition 4.4(b) and (a) that $0\leq S,T\leq 1$ and that $S$ is continuous and $T$ is continuous away from $I\times\{(0,0)\}$. Define functions $H,K\colon I\times I^{2}\to\Delta^{2}$ as follows, where we write $S,T$ instead of $S(p,s,t)$ and $T(p,s,t)$, and $t^{\prime}$ instead of $t^{\prime}(s,t)$

[TABLE]

They are well defined since by Propositions 4.4(b) $0\leq t^{\prime}\leq 1$ and we have seen that $0\leq S,T\leq 1$. They are continuous away from $I\times\{(0,0)\}$ because $S,T$ are and $t^{\prime}$ is continuous away from $(0,0)$. Also, it follows from Proposition 4.4(d) that

[TABLE]

One then checks that

[TABLE]

Inspection of the definition of $H$ and $K$ gives

[TABLE]

Therefore the homotopies $H_{p}$ and $K_{p}$ can be concatenated to form a homotopy $\tilde{\Theta}\colon I\times I^{2}\to\Delta^{2}$ from $\tilde{\alpha}$ to $\tilde{\beta}$ with the properties in the statement of this proposition. ∎

Definition 4.6**.**

Let $A,Y,Z$ be compact Hausdorff spaces and assume that $Y,Z$ are not empty. Let

[TABLE]

be the restriction of the quotient maps (4.1). Since $Y,Z\neq\emptyset$ the second map is a quotient map. Compactness of all spaces implies that $A\times q_{Y,Z}\times I$ in the left vertical map in the diagram below is a quotient map too. It can be described explicitly by the formula

[TABLE]

By Proposition 4.5 and inspection of the formula above, for any $0\leq p\leq 1$ the composition of the top horizontal arrow with the vertical arrow on the right respects the quotient map on the left. We finally define $\Theta\colon(A\times YZ\times I)\times I\to AYZ$ to be the homotopy whose fibres $\Theta_{p}$ are the unique maps which render the following diagram commutative.

[TABLE]

Definition 4.7**.**

Let $\alpha,\beta\colon A\times YZ\times I\to AYZ$ be the maps $\alpha=\theta|_{p=0}$ and $\beta=\theta|_{p=1}$.

Proposition 4.8**.**

The maps $\alpha$ and $\beta$ are homotopic and have the explicit formula

[TABLE]

Proof.

The homotopy is provided by $\theta_{p}$. The formulas are immediate from the explicit description of $\tilde{\alpha}$ and $\tilde{\beta}$ in Definition 4.3 and Proposition 4.5. ∎

The explicit formulas for $\alpha,\beta$ in Proposition 4.8 give the next straightforward calculation.

Proposition 4.9**.**

1. (1)

The restriction $\alpha|_{A\times YZ\times\{0\}}\colon A\times YZ\to AYZ$ is the composition $A\times YZ\xrightarrow{\mathrm{proj}}YZ\xrightarrow{\operatorname{incl}}AYZ$. 2. (2)

The restriction $\alpha|_{A\times YZ\times\{1\}}\colon A\times YZ\to AYZ$ factors through the inclusion $AZ\subseteq AYZ$ and is given by $(a,[sy,(1-s)z])\mapsto[sa,(1-s)z]$. 3. (3)

The restriction $\beta|_{A\times YZ\times\{1\}}\colon A\times YZ\to AYZ$ is the composition $A\times YZ\xrightarrow{\mathrm{proj}}A\xrightarrow{\operatorname{incl}}AYZ$.

The following facts are again straightforward calculations:

Proposition 4.10**.**

Let $A,X,Y$ be compact Hausdorff $G$-spaces, $X,Y\neq\emptyset$. Recall the maps $J_{Z}$ and $\psi$ from (4.3) and Definition 4.2, and let $\pi\colon A\times Y\times I\to AY$ be the restriction of the quotient map (4.1). Then the following diagrams commute.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

To check (4.9) we use the formula for $\alpha$ in Proposition 4.8, for $J_{Z}$ in (4.4) and for $\psi$ in (4.7), to calculate

[TABLE]

The commutativity of (4.10) follows from Proposition 4.9(1) , the naturality of $\psi$ with respect to $A\to*$, and the observation that $J_{Z}$ is $\psi_{*,Y,XY,Z}$.

The commutativity of (4.11) follows by the following calculation which uses Proposition 4.9(2), and equations (4.4) and (4.7).

[TABLE]

Finally, (4.12) follows from (4.6) and Proposition 4.9(3). ∎

5. Filtration of $G$-spaces

Let $G$ be a finite group. Let $(H)$ denote the conjugacy class of $H\leq G$. Enumerate the conjugacy classes of $G$

[TABLE]

so that $|H_{i}|\geq|H_{i+1}|$. In this way, if $H_{i}$ is conjugate to a proper subgroup of $H_{j}$ then $i>j$.

Let $X$ be a $G$-space. Let $G_{x}$ denote the isotropy group of $x\in X$. For any $0\leq q\leq r$ set

[TABLE]

We obtain a filtration of $X$, see [13, Chap. I.6],

[TABLE]

If $X$ is a $G$-CW complex then one checks that $X_{q}$ are subcomplexes [13, Prop. II.1.12]. The assignment $X\mapsto X_{q}$ is a functor giving rise to natural maps

[TABLE]

This is because $f(X_{q})\subseteq Y_{q}$ by the choice of the enumeration (5.1), see [13, I.(6.3)].

Proposition 5.1**.**

Let $X,Y$ be compact Hausdorff $G$-spaces.

1. (1)

If $H\leq G$ then $(XY)^{H}=X^{H}Y^{H}$ (where $XY$ denotes the join). 2. (2)

For any $1\leq q\leq r$

[TABLE] 3. (3)

Set $H=H_{q}$ for some $1\leq q\leq r$. Then

[TABLE]

Proof.

Notice that the isotropy group of $[x,y,t]\in XY$ is $G_{x}$ if $t=1$ and $G_{y}$ if $t=0$ and $G_{x}\cap G_{y}$ if $0<t<1$. From this items (1) and (2) follow easily.

Set $H=H_{q}$ for some $1\leq q\leq r$. By construction

[TABLE]

The choice of the enumeration (5.1) implies item (3) since $x\in X^{H}$ if and only if $H\leq G_{x}$. ∎

Recall that if $X$ is a $G$-space and $H\leq G$ then $X^{H}$ admits an action of $WH=N_{G}H/H$. This gives a functor $X\mapsto X^{H}$ from $G$-spaces to $WH$-spaces. There results a natural map

[TABLE]

Proposition 5.2**.**

Let $X,Y,Z$ be compact Hausdorff $G$-spaces.

1. (1)

The join map $J_{Z}$ (4.3) renders the following square commutative

[TABLE] 2. (2)

Let $H\leq G$. Then

[TABLE]

Proof.

Item (1) follows from Proposition 5.1(1) and by inspection of (4.4). For item (2), suppose that $x\in X^{H}\setminus X_{q-1}^{H}$. Then $H\leq G_{x}$ and by choice of the enumeration (5.1), $G_{x}\in(H_{i})$ for some $i\leq q$. Since $x\notin X_{q-1}$ it follows that $i=q$ and therefore $G_{x}=H$. In particular, $WH_{x}$ is trivial. ∎

Proposition 5.3**.**

Let $X$ be a $G$-CW complex. Set $H=H_{q}\in\operatorname{Iso}_{G}(X)$ for some $1\leq q\leq r$. Then there is a pullback square, natural in both $X$ and $Y$

[TABLE]

whose rows are fibrations, hence the vertical arrows induce homeomorphisms on all fibres.

Proof.

Since $\operatorname{Iso}_{G}(X_{q},X_{q-1})$ is the conjugacy class of $H=H_{q}$, there is a pushout square

[TABLE]

in which the vertical arrows are inclusions of $G$-CW complexes, i.e $G$-cofibrations, and which is is natural in $X$. The pullback square is obtained upon applying the functor $\operatorname{map}^{G}(-,Y)$ to this pushout square and observing that $X_{q}^{H}=X^{H}$ (Proposition 5.1(3)) and that if $A$ is a $WH$-space then there are natural homeomorphisms

[TABLE]

∎

6. The stabilization lemma

The purpose of this section is to prove the following Proposition.

Proposition 6.1**.**

Let $G$ be a finite group. Let $X,Y,Z$ be finite $G$-CW complexes. Let $k\geq 0$. Assume that

1. (1)

$\operatorname{Iso}_{G}(X)=\operatorname{Iso}_{G}(Y)=\operatorname{Iso}_{G}(XYZ)$,

and that for any $H\in\operatorname{Iso}_{G}(X)$

1. (2)

$\dim Y^{H}>k$, 2. (3)

$\dim X^{H}-\dim\left(\bigcup_{K\gneq H}X^{K}\right)>k$. 3. (4)

$\operatorname{conn}(XY)^{H}\geq\dim X^{H}+\dim Y^{H}$* and $\operatorname{conn}(XYZ)^{H}\geq\dim X^{H}+\dim(YZ)^{H}$* 4. (5)

$F(Y^{H},(XY)^{H})\xrightarrow{J_{Z^{H}}}F((YZ)^{H},(XYZ)^{H})$* is a non-equivariant $(\dim X^{H}+k+1)$-equivalence (see (4.3) and Proposition 5.1(1)**).*

Then the natural map

[TABLE]

is a $k$-equivalence.

Proof.

We will use the filtration (5.2) and show that the composition

[TABLE]

is a $k$-equivalence for any $0\leq q\leq r$, where $j$ denotes the inclusion $(XYZ)_{q}\subseteq(XY)_{q}Z$, see Proposition 5.1(2). The claim of the proposition follows for $q=r$.

The proof is by induction on $0\leq q\leq r$. The base of induction $q=0$ is a triviality since $(XY)_{q}=(XYZ)_{q}=\emptyset$. We therefore assume that (6.1) is a $k$-equivalence for $q-1$ and we prove it for $q\leq r$. Set

[TABLE]

Since $X\subseteq XY\subseteq XYZ$, it follows from hypothesis (1) that $\operatorname{Iso}_{G}(X)=\operatorname{Iso}_{G}(XY)=\operatorname{Iso}_{G}(XYZ)$. By the definition of the filtration (5.2), if $H\notin\operatorname{Iso}_{G}(X)$ then $(XY)q=(XY)_{q-1}$ and $(XYZ)_{q}=(XYZ)_{q-1}$, in which case the induction step follows trivially from its hypothesis. We therefore assume that

[TABLE]

Naturality of $J_{Z}$ gives the following commutative diagram whose rows are fibrations and in which $i$ denotes the inclusions $(XY)_{q-1}\subseteq(XY)_{q}$ and $(XYZ)_{q-1}\subseteq(XYZ)_{q}$, and $j$ denotes the inclusions $(XYZ)_{q}\subseteq(XY)_{q}Z$ and $(XYZ)_{q-1}\subseteq(XY)_{q-1}Z$.

[TABLE]

The vertical arrow on the right is a $k$-equivalence by the induction hypothesis, so by Lemma 2.2(1) it remains to show that the fibres of $i^{*}$ are $k$-equivalent.

Proposition 5.1(1) and the formula (4.4) for $J_{Z}(f)$ easily imply the commutativity of the following diagram, where $i$ and $\ell$ denote the inclusions of the $H$-fixed points of $(XY)_{q-1}\subseteq(X_{q-1}Y)_{q}\subseteq(XY)_{q}$ and $(XYZ)_{q-1}\subseteq(X_{q-1}YZ)_{q}\subseteq(XYZ)_{q}$, and $j$ denoted the inclusion of the $H$-fixed points of $(XYZ)_{q-1}\subseteq(XY)_{q-1}Z$.

[TABLE]

By Proposition 5.2(1), the maps $\operatorname{res}_{G}^{H}$ in (5.3) give rise to a natural transformation between the commutative square (6.2) to the square in the middle of (6.3). By Proposition 5.3 the fibres of the rows of (6.2) over any $f\in\operatorname{map}^{G}((XY)_{q-1},XY)$ are homeomorphic to the fibres over $\operatorname{res}_{G}^{H}(f)$ of the rows of the middle square of (6.3).

Therefore, it suffices to prove that the fibres of the rows of the 2nd square in (6.3) are $k$-equivalent for any choice of basepoint in $\operatorname{map}^{WH}((XY)_{q-1}^{H},(XY)^{H})$. If $Z^{H}=\emptyset$ then this is a triviality since $J_{Z^{H}}$ and $j^{*}$ are the identity maps. So for the remainder of the proof we assume that

[TABLE]

It follows from hypothesis (3) and from Proposition 5.1(3) that $\dim X^{H}-\dim X_{q-1}^{H}\geq k+1$. Together with hypotheses (4) and (2) we get

[TABLE]

Similarly,

[TABLE]

Since $H\in\operatorname{Iso}_{G}(X)$ Proposition 5.2(2) implies that $\operatorname{Iso}_{WH}((X_{q-1}Y)^{H},(XY)_{q-1}^{H})=\{e\}$. Corollary 2.5 applies with $(XY)_{q-1}^{H}\subseteq(X_{q-1}Y)^{H}$ and with $(XYZ)_{q-1}^{H}\subseteq(X_{q-1}YZ)^{H}$ to show that both maps $\ell^{*}$ in (6.3) are $k$-equivalences. It follows from Lemma 2.2(2) that the fibres of $i^{*}$ at the top and bottom squares of (6.3) are $k$-equivalent. Since $\ell^{*}$ are bijective on components, it suffices to show that the fibres of $i^{*}$ at the top and bottom of (6.3) are $k$-equivalent via the curved arrows.

This is indeed the case by applying Proposition 6.2 below with $X_{q-1}^{H}\subseteq X^{H}$ and $Y^{H}$ and $Z^{H}$ and $G=WH$. To see this, notice first that $X^{H}$ and $Y^{H}$ are not empty since $H\in\operatorname{Iso}_{G}(X)=\operatorname{Iso}_{G}(Y)$. Also $Z^{H}\neq\emptyset$ by assumption. Hypothesis (1) of Proposition 6.2 follows from Proposition 5.2(2). Hypothesis (2) of Proposition 6.2 is hypothesis (5) of this proposition. Hypothesis (3) of Proposition 6.2 follows from hypotheses (4) and (2) of this proposition. ∎

Proposition 6.2**.**

Let $G$ be a finite group and $X,Y,Z$ be finite non-empty $G$-CW complexes and $X^{\prime}\subseteq X$ a $G$-subcomplex. Let $k\geq 0$. Suppose that

1. (1)

$\operatorname{Iso}_{G}(X,X^{\prime})=\{e\}$. 2. (2)

$F(Y,XY)\xrightarrow{J_{Z}}F(YZ,XYZ)$* is a non-equivariant $(\dim X+k+1)$-equivalence.* 3. (3)

$\operatorname{conn}(XY)\geq\dim X+k+1$.

Then the maps induced on fibres of the horizontal arrows in the following diagram

[TABLE]

are $k$-equivalences for any choice of basepoint in the space at the top right corner.

Proof.

Let $i\colon X^{\prime}\to X$ be the inclusion. Define the following objects in $\mathfrak{S}$, see Definition 3.1.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The commutativity of these squares is a direct consequence of the naturality of $i\mapsto i^{*}$. The plan of the proof is as follows.

• (a)

Define morphisms $\mathcal{A}\xrightarrow{\Phi}\mathcal{B}\xrightarrow{\Pi}\mathcal{D}$ and $\mathcal{A}\xrightarrow{\Pi}\mathcal{C}\xrightarrow{\Psi}\mathcal{D}$ in $\mathfrak{S}$. Note: We used $\Pi$ to denote two different morphisms; This will create no source of confusion and the reason for the choice will become apparent in (6.5) and (6.6) where they are defined.

• (b)

Show that both $\Lambda(\mathcal{B},\underline{y})\xrightarrow{\Lambda(\Pi)}\Lambda(\mathcal{D},\Pi(\underline{y}))$ and $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\Pi)}\Lambda(\mathcal{C},\Pi(\underline{x}))$ are weak homotopy equivalences for any choice of basepoints $\underline{x}$ for $\mathcal{A}$ and $\underline{y}$ for $\mathcal{B}$, see Definition 3.1 and equation (3.2).

• (c)

Show that $\Lambda(\mathcal{C},\underline{y})\xrightarrow{\Lambda(\Psi)}\Lambda(\mathcal{D},\Psi(\underline{y}))$ is a $k$-equivalence for any choice of basepoint $\underline{y}$ in $\mathcal{C}$.

• (d)

Show that $\mathcal{A}\xrightarrow{\Pi\circ\Phi}\mathcal{D}$ is homotopic to $\mathcal{A}\xrightarrow{\Psi\circ\Pi}\mathcal{D}$ (Definition 3.3).

• (e)

Deduce that $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\Phi)}\Lambda(\mathcal{B},\Phi(\underline{x}))$ is a $k$-equivalence for any choice of base point $\underline{x}$ for $\mathcal{A}$. Use this and Lemma 3.2(3) to complete the proof.

With the indexing in (3.1), we will now describe maps $\Phi_{i}\colon\mathcal{A}_{i}\to\mathcal{B}_{i}$ and $\Pi_{i}\colon\mathcal{B}_{i}\to\mathcal{D}_{i}$ and $\Pi_{i}\colon\mathcal{A}_{i}\to\mathcal{C}_{i}$ and $\Psi_{i}\colon\mathcal{C}_{i}\to\mathcal{D}_{i}$, where $i=0,\dots,3$. We will then show that these are the components of natural transformations $\mathcal{A}\xrightarrow{\Phi}\mathcal{B}$ and $\mathcal{B}\xrightarrow{\Pi}\mathcal{D}$ and $\mathcal{A}\xrightarrow{\Pi}\mathcal{C}$ and $\mathcal{C}\xrightarrow{\Psi}\mathcal{D}$.

Notation: In what follows we will use the letter $A$ to represents either $X$ or $X^{\prime}$,

[TABLE]

Define $\Phi_{i}\colon\mathcal{A}_{i}\to\mathcal{B}_{i}$ using the maps $J_{Z}$ (4.3) and the inclusion $\operatorname{incl}\colon XY\to XYZ$ as follows,

[TABLE]

Let $\pi\colon A\times YZ\times I\to AYZ$ be the map in (4.2). Let $\pi_{A}\colon A\times YZ\to A$ and $\pi_{YZ}\colon A\times YZ\to YZ$ be the projections. Define $\Pi_{i}\colon\mathcal{B}_{i}\to\mathcal{D}_{i}$ as follows,

[TABLE]

Let $\pi\colon A\times Y\times I\to AY$ be as in (4.2) and $\pi_{A}\colon A\times Y\to A$ and $\pi_{Y}\colon A\times Y\to Y$ be the projections. Set $\Pi_{i}\colon\mathcal{A}_{i}\to C_{i}$ as follows

[TABLE]

Use the maps $\psi_{A\times I,Y,XY,Z}$ and $\psi_{A,Y,XY,Z}$ in Definition 4.2 to define $\Psi_{i}\colon\mathcal{C}_{i}\to\mathcal{D}_{i}$

[TABLE]

Claim 1: The maps $\Phi_{0},\dots,\Phi_{3}$ in (6.4) define a natural transformation $\Phi\colon\mathcal{A}\to\mathcal{B}$.

Proof: With the indexing of Definition 3.1, the naturality of $J_{Z}$ and the equality $\operatorname{incl}_{*}\circ i^{*}=i^{*}\circ\operatorname{incl}_{*}$ imply that $\Phi_{1}\circ a_{31}=b_{31}\circ\Phi_{3}$ and $\Phi_{0}\circ a_{20}=b_{20}\circ\Phi_{2}$. The commutative square in (4.6) implies the commutativity of the following diagram

[TABLE]

Composing the 2nd factor of the 2nd column with $\operatorname{map}^{G}(AZ,XYZ)\xrightarrow{i_{A}^{*}}\operatorname{map}^{G}(A,XYZ)$ and using the commutative triangle in (4.6), it follows that $\Phi_{2}\circ a_{32}=b_{32}\circ\Phi_{3}$ and $\Phi_{0}\circ a_{10}=b_{10}\circ\Phi_{1}$. Hence, $\Phi\colon\mathcal{A}\to\mathcal{B}$ is a morphism in $\mathfrak{S}$. QED

Claim 2: The maps $\Psi_{0},\dots,\Psi_{3}$ in (6.7) define a morphism $\Psi\colon\mathcal{C}\to\mathcal{D}$.

Proof: This is immediate from the naturality of $\psi$ with respect to the inclusions $X^{\prime}\subseteq X$ and $X^{\prime}\times I\subseteq X\times I$ and the inclusions $A\coprod A\subseteq A\times I$. QED.

Claim 3: The maps $\Pi_{0},\dots,\Pi_{3}$ in (6.6) define a morphism $\Pi\colon\mathcal{A}\to\mathcal{C}$ in $\mathfrak{S}$. Moreover $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\Pi)}\Lambda(\mathcal{C},\Pi(\underline{x}))$ is a weak homotopy equivalence for any basepoint $\underline{x}$ for $\mathcal{A}$. Similarly, the maps $\Pi_{0},\dots,\Pi_{3}$ in (6.5) define a morphism $\Pi\colon\mathcal{B}\to\mathcal{D}$ in $\mathfrak{S}$ and $\Lambda(\mathcal{B},\underline{y})\xrightarrow{\Lambda(\Pi)}\Lambda(\mathcal{D},\Pi(\underline{y}))$ is a weak homotopy equivalence for any basepoint $\underline{y}$ in $\mathcal{B}$.

Proof: We will prove the statements about the maps in (6.6) and $\Pi\colon\mathcal{A}\to\mathcal{C}$. The proof for the maps in (6.5) and $\Pi\colon\mathcal{B}\to\mathcal{D}$ is obtained by replacing $Y$ with $YZ$ everywhere and $\mathcal{A}$ with $\mathcal{B}$ and $\mathcal{C}$ with $\mathcal{D}$.

By applying $\operatorname{map}^{G}(-,XY)$ to the commutative squares

[TABLE]

it follows that $c_{31}\circ\Pi_{3}=\Pi_{1}\circ a_{31}$ and $c_{20}\circ\Pi_{2}=\Pi_{0}\circ a_{20}$. By applying $\operatorname{map}^{G}(-,XY)$ to the pushout square (4.2) we obtain pullback squares

[TABLE]

which are in particular commutative. Thus, $\Pi_{0},\dots,\Pi_{3}$ define a natural transformation $\Pi\colon\mathcal{A}\to\mathcal{C}$. Since the horizontal arrows $a_{32},a_{10}$ and $c_{32}$ and $c_{10}$ are fibrations, the two squares above are homotopy pullback squares and Lemma 3.2(1) shows that $\Lambda(\Pi,\underline{x})$ are weak homotopy equivalences for all basepoints $\underline{x}$ of $\mathcal{A}$. QED.

Claims 1–3 complete steps (a) and (b) in our plan of the proof.

Claim 4: $\Lambda(\mathcal{C},\underline{y})\xrightarrow{\Lambda(\Psi)}\Lambda(\mathcal{D},\Psi(\underline{y}))$ is a $k$-equivalence for any choice of basepoint $\underline{y}$ in $\mathcal{C}$.

Proof: Hypotheses (1) and (2) allow us to apply Lemma 2.4 to the following commutative squares

[TABLE]

[TABLE]

It follows that the fibres of the rows in the $1$st square are $k$-equivalent and those in the $2$nd square are $(k+1)$-equivalent. By construction of the maps $\Psi_{0},\dots,\Psi_{3}$ in (6.7) and Definition 4.2 of the maps $\psi$, it follows that the maps induced on the homotopy fibres in the following diagrams

[TABLE]

are $(k+1)$-equivalences for any choice of basepoints $y_{0}\in\mathcal{C}_{0}$ and $y_{1}\in\mathcal{C}_{1}$. Lemma 3.2(2) shows that $\Lambda(\mathcal{C},\underline{y})\xrightarrow{\Lambda(\Psi)}\Lambda(\mathcal{D},\Psi(\underline{y}))$ is a $k$-equivalence for any basepoint $\underline{y}$ of $\mathcal{C}$. QED.

This completes step (c) of the proof. We turn to the technical proof of step (d).

Claim 5: The morphisms $\mathcal{A}\xrightarrow{\Phi}\mathcal{B}\xrightarrow{\Pi}\mathcal{D}$ and $\mathcal{A}\xrightarrow{\Pi}\mathcal{C}\xrightarrow{\Psi}\mathcal{D}$ are homotopic (Definition 3.3).

Proof: The plan is define an object $\mathcal{H}\in\mathfrak{S}$, a morphism $\mathcal{A}\xrightarrow{\Upsilon}\mathcal{H}$ and a homotopy $\mathcal{H}\xrightarrow{\Xi_{p}}\mathcal{D}$ parameterized by $0\leq p\leq 1$, such that $(\Xi\circ\Upsilon)|_{p=0}=\Pi\circ\Phi$ and $(\Xi\circ\Upsilon)|_{p=1}=\Psi\circ\Pi$.

By applying the functor $\operatorname{map}^{G}(-,XYZ)$ to the commutative square of inclusions

[TABLE]

we obtain the following object $\mathcal{H}$ in $\mathfrak{S}$.

[TABLE]

Define maps $\Upsilon_{i}\colon\mathcal{A}_{i}\to\mathcal{H}_{i}$ ($i=0,\dots,3$) as follows (we use the indexing as in Definition 3.1).

[TABLE]

They give rise to a morphism $\Upsilon\colon\mathcal{A}\to\mathcal{H}$ in $\mathfrak{S}$ by commutativity of the square in (4.6).

We now define homotopies $(\Xi_{i})_{p}\colon\mathcal{H}_{i}\to\mathcal{D}_{i}$ parameterized by $0\leq p\leq 1$. Apply $\operatorname{map}^{G}(-,XYZ)$ to the homotopies $\Theta_{p}\colon A\times YZ\times I\to AYZ$ in Definition 4.6 to obtain the maps

[TABLE]

Definition 4.6 and Proposition 4.5 show that $\Theta_{p}|_{t=0}\overset{\text{def}}{=}\Theta_{p}|_{A\times YZ\times\{0\}}$ factors through the inclusion $YZ\subseteq AYZ$ and that $\Theta_{p}|_{t=1}\overset{\text{def}}{=}\Theta_{p}|_{A\times YZ\times\{1\}}$ factors through the inclusion $AZ\subseteq AYZ$. By applying $\operatorname{map}^{G}(-,XYZ)$ to these square homotopies we obtain the maps

[TABLE]

Naturality of the construction of $\Theta$ with respect to the inclusion $X^{\prime}\subseteq X$ implies the commutativity of the squares

[TABLE]

Furthermore, by applying $\operatorname{map}^{G}(-,XYZ)$ to the commutative square

[TABLE]

we obtain the commutativity of

[TABLE]

Therefore $\Xi_{0},\dots,\Xi_{3}$ give rise to a homotopy $\Xi\colon I\times\mathcal{H}\to\mathcal{D}$ in $\mathfrak{S}$. Composition with $\Upsilon\colon\mathcal{A}\to\mathcal{H}$ gives a homotopy $\Xi\circ\Upsilon\colon I\times\mathcal{A}\to\mathcal{D}$ parameterized by $p\in I$.

It remains to show that $(\Xi\circ\Upsilon)|_{p=0}=\Psi\circ\Pi$ and that $(\Xi\circ\Upsilon)|_{p=1}=\Pi\circ\Phi$. We start with $p=1$. By Proposition 4.8, $\Theta|_{p=1}=\beta$ is the natural map $A\times YZ\times I\xrightarrow{\pi}A(YZ)\cong AYZ$ in (4.2). By the definition of $\Phi_{1},\Phi_{3}$ in (6.4) and $\Pi_{1},\Pi_{3}$ in (6.5) and $\Upsilon_{1},\Upsilon_{3}$ in (6) and $\Xi_{1},\Xi_{3}$ in (6.9), it follows that for $i=1,3$

[TABLE]

Proposition 4.8 shows that $\Theta_{p=1}|_{t=0}=\beta|_{A\times YZ\times\{0\}}$ is the projection $A\times YZ\to YZ$ and that $\Theta_{p=1}|_{t=1}=\beta|_{A\times YZ\times\{1\}}$ is the composition of the projection $A\times YZ\to A$ followed by the inclusion into $AZ$. Since $(\Xi_{0})_{p}$ and $(\Xi_{2})_{p}$ are obtained by applying $\operatorname{map}^{G}(-,XYZ)$ to the first column of (6.11), the commutative triangle in (4.6) together with (6.4) and (6.5) show that for $i=0,2$

[TABLE]

It follows that

[TABLE]

It remains to show that $(\Xi\circ\Upsilon)|_{p=0}=\Psi\circ\Pi$. First, we claim that for $i=1,3$

[TABLE]

The first equality follows from the definitions of $\Xi_{i}$ and $\Upsilon_{i}$ in (6.9) and (6), where $i=1,3$, and from Definition 4.7; The second equality follows from the commutative square (4.9) in Proposition 4.10, and the third from the definitions of $\Psi_{i}$ and $\Pi_{i}$ in (6.7) and (6.6).

Let $i=0,2$. We claim that

[TABLE]

The first equality follows from the definition of $\Upsilon_{i}$ and $\Xi_{i}$ in (6) and (6.9); The second follows from (4.10) and (4.11) in Proposition 4.10, and the third from the definition of $\Psi_{i}$ and $\Pi_{i}$ in (6.7) and (6.6). It follows that $(\Xi\circ\Upsilon)|_{p=0}=\Psi\circ\Pi$. Q.E.D

This completes step (d) of the proof. We are now ready to complete the proof of the proposition as outlined in step (e).

Claims 3 and 4 and the functoriality of $\Lambda$ imply that $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\Psi\circ\Pi)}\Lambda(\mathcal{D},\Psi\circ\Pi(\underline{x}))$ is a $k$-equivalence for any choice of basepoint $\underline{x}$ in $\mathcal{A}$. Claim 5 together with Lemma 3.4 show that $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\Pi\circ\Phi)}\Lambda(\mathcal{D},\Pi\Phi(\underline{x}))$ is a $k$-equivalence. From Claim 3 and the functoriality of $\Lambda$ we deduce that $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\Phi)}\Lambda(\mathcal{B},\Phi(\underline{x}))$ is a $k$-equivalence for any basepoint $\underline{x}$ in $\mathcal{A}$.

By hypothesis (3), $\operatorname{conn}(XY)\geq\dim X+k+1$ and therefore also $\operatorname{conn}(XYZ)\geq\dim X+k+1$, see for example [8, Lemma 2.3]. Thanks to hypothesis (1), we may apply Corollary 2.5 to $X^{\prime}\subseteq X$ and to $XY$ and $XYZ$ and deduce, in light of the definitions of $\mathcal{A}$ and $\mathcal{B}$, that the horizontal arrows in the following square are $(k+1)$-equivalences.

[TABLE]

In particular, their fibres are $(k+1)$-connected, and therefore $\Phi_{2}$ and $\Phi_{0}$ induce $k$-equivalences among them. We have already seen that $\Lambda(\mathcal{A},\underline{x})\xrightarrow{\Lambda(\Phi)}\Lambda(\mathcal{B},\Phi(\underline{x}))$ are $k$-equivalences, therefore we may apply Lemma 3.2(3) to deduce that in the commutative square

[TABLE]

the vertical arrows induce $k$-equivalences on all the fibres of $a_{31}$ and $b_{31}$. Given the definition of $\Phi_{1}$ and $\Phi_{3}$ in (6.4), this is exactly the claim of this proposition. ∎

7. The limit groups

The purpose of this section is to prove Proposition 7.3 below. It will make an essential use of equivariant stable homotopy theory. Our main reference is Lewis-May-Steinberg [6]. Let $G$ be a finite group. We will fix a universe $\mathcal{U}$ for the representations of $G$ and an indexing set $\mathcal{A}$ for $\mathcal{U}$ [6, Definition 2.1]. A spectrum $E$ is a collection of pointed $G$-spaces $E(V)$ for every $V\subseteq\mathcal{A}$, subject to some conditions [6, Def. 2.1]. There is a functor $\Sigma^{\infty}$ from the category of pointed $G$-spaces to the category of $G$-spectra. It has a right adjoint $\Omega^{\infty}$ which assigns to a spectrum $E$ its zeroth space [6, Proposition 2.3]. We denote, as usual, $Q=\Omega^{\infty}\circ\Sigma^{\infty}$ [6, p. 14]. If $A$ is a pointed $G$-space, the function spectrum $F(A,E)$ is obtained by applying $F(A,-)$ to each space of $E$ [6, Def. 3.2]. If $H\leq G$, there is a fixed point spectrum $E^{H}$ which is obtained, once again, by taking $H$-fixed points of the spaces of $E$ [6, Def. 3.7]. It has a structure of a $WH$-spectrum where $WH=N_{G}H/H$.

The Borel construction of a $G$-space $X$ is the orbit space $EG\times_{G}X=(EG\times X)/G$. Let $X$ be a pointed $G$-space. Denote $EG_{+}\wedge_{G}X\overset{\text{def}}{=}(EG_{+}\wedge X)/G$. The following important result [6, Section V.11] gives a complete description of the fixed point spectrum $(\Sigma^{\infty}X)^{G}$

[TABLE]

where $H$ runs through representatives of the conjugacy classes of the subgroups of $G$ and where $WH=N_{G}H/H$ acts in the natural way on $X^{H}$.

Let $U$ be a real representation of $G$. Let $S^{U}$ denote the one point compactification of $U$ with basepoint $\alpha_{1}=\infty$. Clearly $\alpha_{0}=0\in U$ is also fixed by $G$. Notice that for any $H\leq G$ we have $(S^{U})^{H}=S^{V}$ where $V=U^{H}$.

Let $C(f)$ denote the mapping cone of a map $f\colon A\to B$ of unpointed $G$-spaces; it is equipped with a natural basepoint (the “tip of the cone”). If $A\subseteq B$ we write $C(B,A)$ for the mapping cone of the inclusion.

For a $G$-space $X$ with fixed points $x_{0},x_{1}$, denote by $\mathbb{P}_{x_{0},x_{1}}X$ the space of paths $\omega\colon I\to X$ with $\omega(0)=x_{0}$ and $\omega(1)=x_{1}$. It has a natural action of $G$.

Lemma 7.1**.**

1. (1)

Let $X$ be a pointed finite CW-complex such that $\tilde{H}_{i}(X)=0$ for all $0\leq i\leq m$. Then $\pi_{i}\Sigma^{\infty}X=0$ for all $0\leq i\leq m$. 2. (2)

Let $X$ be a finite $G$-CW complex such that $\tilde{H}_{i}(X)=0$ for all $0\leq i\leq m$. Then $\pi_{i}\Sigma^{\infty}(EG_{+}\wedge_{G}X)=0$ for all $0\leq i\leq m$.

Proof.

(1). This follows from Atiyah-Hirzebruch spectral sequence $\tilde{H}_{i}(X,E_{j}(*))\Rightarrow\tilde{E}_{i+j}(X)$ applied to the sphere spectrum $E=\mathbf{S}$.

(2). There is a $G$-cofibre sequence where $EG$ retracts off $EG\times X$ equivariantly (via the basepoint of $X$).

[TABLE]

By taking $G$-orbits we get a cofibre sequence

[TABLE]

with $BG$ retracting off $X_{hG}=EG\times_{G}X$. The Serre spectral sequence $H_{i}(BG,H_{j}(X))\Rightarrow H_{i+j}(X_{hG})$ of the fibration $X_{hG}\to BG$ shows that $H_{i}(BG)\to H_{i}(X_{hG})$ is an isomorphism for all $0\leq i\leq m$ and therefore $\tilde{H}_{i}(EG_{+}\wedge_{G}X)=0$ for all $0\leq i\leq m$. The result follows from item (1). ∎

Lemma 7.2**.**

Let $V$ be a $G$-representation and $X\subseteq V$ a finite $G$-CW complex. Set $n=\dim V$. Then there are isomorphisms for all $0\leq i\leq n-2$

[TABLE]

Proof.

Lemma 7.1(2) shows that $\pi_{i}\Sigma^{\infty}(EG_{+}\wedge_{G}S^{V})=0$ for $0\leq i\leq n-1$. The long exact sequence in stable homotopy groups of the cofibration $EG_{+}\wedge_{G}(S^{V}\setminus X)\to EG_{+}\wedge_{G}S^{V}\to EG_{+}\wedge_{G}C(S^{V},S^{V}\setminus X)$ gives the result. ∎

Proposition 7.3**.**

Let $U$ be a representation of $G$ and let $X\subseteq U$ be a finite $G$-CW complex. Let $k\geq 0$. Assume that for any $H\in\operatorname{Iso}_{G}(X)$

1. (a)

$\dim X^{H}<\dim U^{H}$. 2. (b)

$\dim X^{H}-\dim\left(\bigcup_{K\gneq H}X^{K}\right)\ >\ k+1$. 3. (c)

$(S^{U}\setminus X)^{H}$* is $WH$-equivariantly homotopy equivalent to a $WH$-CW complex.*

Then $\operatorname{map}^{G}(X,S^{U})$ is path connected and for all $1\leq i\leq k+1$

[TABLE]

If in addition

1. (d)

there exists a $G$-map $X\xrightarrow{\eta}\mathbb{P}_{\alpha_{0},\alpha_{1}}S^{U}$ such that $X^{H}\xrightarrow{\eta}\mathbb{P}_{\alpha_{0},\alpha_{1}}(S^{U})^{H}$ is a $(\dim X^{H}+k)$-equivalence for any $H\in\operatorname{Iso}_{G}(X)$

then for any basepoint $f\in\operatorname{map}^{G}(X,X)$ and every $0\leq i\leq k$ there are isomorphisms (bijection for $i=0$):

[TABLE]

Proof.

We will prove by induction on the filtration $\{X_{q}\}_{q=0}^{r}$ of $X$ in (5.2) that $\operatorname{map}^{G}(X_{q},S^{U})$ is path connected and that there are isomorphisms for all $1\leq i\leq k+1$

[TABLE]

The base of induction is a triviality since $X_{0}=\emptyset$. Assume that (7.3) holds for $q-1$ and we prove it for $1\leq q\leq r$. If $H_{q}\notin\operatorname{Iso}_{G}(X)$ then $X_{q}=X_{q-1}$ and the induction step is trivial. So we assume that $H=H_{q}$ is in $\operatorname{Iso}_{G}(X)$.

Choose some basepoint $f\in\operatorname{map}^{G}(X_{q},S^{U})$. We obtain a fibre sequence (over $f|_{X_{q-1}}$)

[TABLE]

The hypotheses imply that

[TABLE]

We can apply Corollary 2.5 (with $Y=S^{U}$ and $k=0$) to deduce that $j^{*}$ is bijective on components and that $\pi_{0}F=*$. Together with the induction hypothesis on $\operatorname{map}^{G}(X_{q-1},S^{U})$, it follows that $\operatorname{map}^{G}(X_{q},S^{U})$ is path connected, as needed. Therefore we may assume that the basepoint $f$ is the null map.

Since $\operatorname{Iso}_{G}(X_{q})=\operatorname{Iso}_{G}(X_{q-1})\cup(H)$, in order to complete the induction step for (7.3) it remains to show that for every $1\leq i\leq k+1$

(i) $\pi_{i}F\to\pi_{i}\operatorname{map}^{G}(X_{q},S^{U})$ is split injective , and

(ii) $\pi_{i}F\cong\pi_{i}\Sigma^{\infty}\left(EWH_{+}\wedge_{WH}C(S^{U},S^{U}\setminus X)^{H}\right)$.

For the rest of the proof set $V=U^{H}$ and $n=\dim V$. Proposition 5.3 yields the following morphism of fibrations which induces a homeomorphism on the fibres (over the null maps)

[TABLE]

Therefore, we will be finished if we prove (ii) and that $\pi_{i}F\to\pi_{i}\operatorname{map}^{WH}(X_{q-1}^{H},S^{V})$ is split injective for all $1\leq i\leq k+1$.

Since $QS^{V}=\operatorname{colim}_{V^{\prime}}\Omega^{V^{\prime}}\Sigma^{V^{\prime}}S^{V}$ [6, p. 14], Freudenthal’s theorem implies that the natural map $S^{V}\to QS^{V}$ is a non-equivariant $(2n-2)$-equivalence. Hypotheses (a) and (b) imply that

[TABLE]

because $\dim\,(\cup_{K\gneq H}X^{K})\geq-1$. In particular

[TABLE]

Application of Proposition 5.2(2) and Lemma 2.4 to $X_{q-1}^{H}\subseteq X^{H}$ and $S^{V}\to QS^{V}$ shows that in the commutative diagram

[TABLE]

the map $F\to F^{\prime}$ between the fibres (over the null maps) is a $(k+1)$-equivalence. Thus, to complete the induction step of (7.3) it suffices to prove that for every $1\leq i\leq k+1$

(i’) $\pi_{i}F^{\prime}\to\pi_{i}\operatorname{map}^{WH}(X^{H},QS^{V})$ is split injective, and

(ii’) $\pi_{i}F^{\prime}$ is isomorphic to the groups in (ii).

Let $A$ be a pointed $G$-CW complex. The definitions of $\Omega^{\infty}$ and $Q=\Omega^{\infty}\Sigma^{\infty}$ [6, p. 14], of the function spectra $F(A,E)$ [6, Prop. 3.6], and of fixed point spectra [6, Def. 3.7], imply that there are natural homeomorphisms

[TABLE]

Therefore the fibration $\ell^{*}$ in the 2nd row of (7.4) is obtained by applying the functor $\Omega^{\infty}$ and $WH$-fixed points to the morphism of $WH$-spectra

[TABLE]

We will now exploit $V$-duality [6, Chap. III]. Recall that $C(X,\emptyset)=X_{+}$ where $X$ is an unpointed space [6, page 142]. The definition of $V$-duality [6, Defn. 3.4] together with the formula for the map $\epsilon/(?)$ [6, Prop. 3.1] and the construction of $V$-duality for compact $G$-ENRs [6, Construction 4.5, page 145], give rise to the following homotopy commutative diagram of $WH$-spectra

[TABLE]

By definition of $V$-duality (or by construction), the vertical arrows in this diagram are weak equivalences of $WH$-spectra [6, Def. 4.4]. By applying the fixed points functor $(-)^{WH}$, and the functor $\Omega^{\infty}$ we see that in order to prove (i’) and (ii’) it suffices to prove that

(i”) $\pi_{i}\left(\Sigma^{\infty}C(S^{V},S^{V}\setminus X^{H})\right)^{WH}\xrightarrow{\pi_{i}(\ell^{*})}\pi_{i}\left(\Sigma^{\infty}C(S^{V},S^{V}\setminus X_{q-1}^{H})\right)^{WH}$ is split surjective for all $1\leq i\leq k+1$ and surjective for $i=k+2$, and

(ii”) the kernels of the homomorphisms in (i”) are isomorphic to the groups in (ii) for all $1\leq i\leq k+1$.

On the level of spectra, [6, Section V.11] quoted above shows that the map in (i”) is induced by the map of spectra

[TABLE]

where the sum is over all the conjugacy classes of subgroups $K\leq WH$ and by $WK$ we mean $N_{WH}(K)/K$. Any $K\leq WH$ has the form $K=L/H$ for some $H\leq L\leq N_{G}H$. If $K\neq 1$ then $L\supsetneq H$ and in this case $X^{L}=X_{q-1}^{L}$ and it follows that the maps of the summands corresponding to $K\neq 1$ are equivalences. It remains to examine the summand $K=1$, namely the map

[TABLE]

The hypotheses and Proposition 5.1(3) show that

[TABLE]

In particular $H^{i}(X_{q-1}^{H})=0$ for all $i\geq n-k-2$. Alexander duality implies that $\tilde{H}_{i}(S^{V}\setminus X_{q-1}^{H})=0$ for all $0\leq i\leq k+1$. Also, $\tilde{H}_{i}(S^{V})=0$ for all $0\leq i\leq k+1$ since $\operatorname{conn}(S^{V})=n-1\geq\dim X^{H}\geq k+1$. There is a cofibre sequence

[TABLE]

The long exact sequence in stable homotopy groups together with Lemma 7.1(2) show that $\pi_{i}$ of the right hand side of (7.5) vanishes for $0\leq i\leq k+1$ and that $\pi_{k+2}\Sigma^{\infty}\gamma$ is surjective. Also $\pi_{k+2}$ of (7.5) is surjective because $\pi_{k+2}\Sigma^{\infty}\gamma$ factors through it. In particular (i”) and (ii”) follow and the induction step is complete.

Let $X\xrightarrow{\eta}\mathbb{P}_{\alpha_{0},\alpha_{1}}S^{U}$ be as in hypothesis (d). Applying Lemma 2.4 with $\emptyset\subseteq X$ and with $\eta$ shows that $\operatorname{map}^{G}(X,X)\to\operatorname{map}^{G}(X,\mathbb{P}_{\alpha_{0},\alpha_{1}}S^{U})$ is a $k$-equivalence. By inspection, and since we have shown that $\operatorname{map}^{G}(X,S^{U})$ is path connected,

[TABLE]

We have seen that if $H\in\operatorname{Iso}_{G}(X)$ and $n=\dim U^{H}$ then $n-1\geq\dim X^{H}\geq k+1$. Lemmas 7.2 and 7.1(2) apply to show that for $0\leq i\leq k$ there are isomorphisms (bijection if $i=0$)

[TABLE]

∎

8. Proof of the main theorems

In this section we fix a finite group $G$ and a sequence of representation $U_{1},U_{2},\dots$ satisfying hypothesis (U) in Section 1. For any $m\leq n$ we write

[TABLE]

Recall that $\mathcal{F}(U_{\bullet})$ is the smallest collection of subgroups of $G$ which contains $\operatorname{Iso}_{G}(S(V))$ for all $V\in\operatorname{Irr}(U_{\bullet})$ and is closed to intersection of groups. Illman showed in [4] that $S(V)$ is a finite $G$-CW complex for any representation $V$. Also $S(V)^{H}=S(V^{H})$ is a sphere.

Lemma 8.1**.**

Fix $m\geq 1$ and $k\geq 0$. Then for every sufficiently large $n$

1. (1)

$\operatorname{Iso}_{G}(S(U_{m\leq\bullet\leq n}))=\mathcal{F}(U_{\bullet})$. 2. (2)

$\dim S(U_{m\leq\bullet\leq n})^{H}\geq k$* for every $H\in\mathcal{F}(U_{\bullet})$.* 3. (3)

$\dim S(U_{m\leq\bullet\leq n})^{H}-\dim\bigcup_{K\gneq H}S(U_{m\leq\bullet\leq n})^{K}\,\geq\,k$* for every $H\in\mathcal{F}(U_{\bullet})$. (In this inequality the dimension of the empty set is $-1$).*

Proof.

By construction of the join

[TABLE]

Thus, $\operatorname{Iso}_{G}(S(U_{\leq 1}))\subseteq\operatorname{Iso}_{G}(S(U_{\leq 2}))\subseteq\cdots$. Since $G$ is finite this sequence stabilizes on a collection $\mathcal{F}$. It also follows that $\mathcal{F}=\operatorname{Iso}_{G}(S(V))$ for some representation with irreducible summands in $\operatorname{Irr}(U_{\bullet})$. Since hypothesis (U) is in force, it is clear that $\mathcal{F}=\mathcal{F}(U_{\bullet})$.

If $H\in\mathcal{F}$ then $\dim V^{H}\geq 1$ and moreover, if $K\gneq H$ then $\dim V^{H}>\dim V^{K}$. Hypothesis (U) implies that if $n$ is large enough then $U_{m\leq\bullet\leq n}$ contains $\oplus_{k+1}V$ and the lemma follows (inequality (3) needs to be proven separately for $H\lneq G$ and $H=G$). ∎

Proof of Theorem 1.1.

Proof of the stabilization: Fix $k\geq 0$. By Lemma 8.1 there exist integers $n_{0}\geq m\geq 1$ such that $X\overset{\text{def}}{=}S(U_{\leq m})$ and $Y^{\prime}\overset{\text{def}}{=}S(U_{m+1\leq\bullet\leq n_{0}})$ satisfy

[TABLE]

and for any $H\in\mathcal{F}(U_{\bullet})$

[TABLE]

Let $n\geq n_{0}$ and set $Y=S(U_{m+1\leq\bullet\leq n})$ and $Z=S(U_{n+1})$. Then, $S(U_{\leq n})\cong X*Y$ and $S(U_{\leq n+1})\cong X*Y*Z$ and the first statement of the theorem is that

[TABLE]

is a $k$-equivalence. To prove this we apply Proposition 6.1. First, $X,Y$ and $Z$ are finite $G$-CW complexes by Illman’s result [4]. Since $Y^{\prime}\subseteq Y$, the choice of $m$ and $n_{0}$ guarantees that

[TABLE]

so hypothesis (1) of Proposition 6.1 holds.

Hypothesis (2) also holds since $\dim Y^{H}\geq\dim Y^{\prime}{}^{H}\geq k+1$ for all $H\in\mathcal{F}$, and our choice of $X$ satisfies hypothesis (3). Now, $(X*Y)^{H}\cong S(U_{\leq n})^{H}$ is itself a linear sphere of dimension $\dim X^{H}+\dim Y^{H}+1$, hence it is a $(\dim X^{H}+\dim Y^{H})$-connected space. Similarly $(X*Y*Z)^{H}$ is a linear sphere of dimension $\dim X^{H}+\dim(Y*Z)^{H}+1$ and hypothesis (4) also holds.

Now, $Z^{H}$ is a sphere and the join of a space $A$ with $S^{m}\cong S^{0}*\dots*S^{0}$ is homeomorphic to the $(m+1)$-fold unreduced suspension of $A$. An iterated use of Proposition 2.7(b) shows that the map $F(X^{H},(X*Y)^{H})\to F(X^{H}*Z^{H},(X*Y)^{H}*Z^{H})$, where $F(-,-)$ denotes the space of (non-equivariant) continuous maps, is a $(\dim(X*Y)^{H}-1)$-equivalence. Hypothesis (5) of Proposition 6.1 holds since $\dim(X*Y)^{H}-1=\dim X^{H}+\dim Y^{H}\geq\dim X^{H}+k+1$.

Calculation of the limit groups: Given $n\geq 1$ write $U=U_{\leq n}$ and $X=S(U)\subseteq U\subseteq S^{U}$. Lemma 8.1 guarantees that if $n$ is large enough then $\operatorname{Iso}_{G}(X)=\mathcal{F}(U_{\bullet})$ and $\dim X^{H}\geq k+2$ and $\dim X^{H}-\dim\cup_{K\gneq H}X^{K}\geq k+2$ for all $H\in\mathcal{F}(U_{\bullet})$. We will show that $\pi_{i}\operatorname{map}^{G}(X,X)$ are isomorphic to the groups in the statement of the theorem for $0\leq i\leq k$.

This follows from Proposition 7.3 which we proceed to check its hypotheses (a)–(d). Clearly, $\dim X=\dim U-1$ which is hypothesis (a). Hypothesis (b) holds by the choice of $n$, and (c) since $S^{U}\setminus S(U)$ is $G$-equivalent to $S^{0}$. For hypothesis (d) let $\eta_{S(U)}\colon S(U)\to S^{U}$ be the map in 2.6 followed by the homeomorphism $\Sigma S(U)\cong S^{U}$. By Proposition 2.7(a) $(\eta_{S(U)})^{H}=\eta_{S(U^{H})}$ is a $(2\dim S(U)^{H}-2)$-equivalence for any $H\in\operatorname{Iso}_{G}(S(U))$, and by the choice of $U$ we have $2\dim S(U)^{H}-2\geq\dim S(U)^{H}+k$. ∎

The “distinguished” fixed points $0,\infty\in S^{U}$ correspond, under the homeomorphism $S^{U}\cong\Sigma S(U)$, to $\alpha_{0},\alpha_{1}\in\Sigma S(U)$ in 2.6. Write $A=\{\alpha_{1}\}$ and $B=\{\alpha_{0},\alpha_{1}\}$. Let

[TABLE]

be the spaces of maps $f\colon S^{U}\to S^{U}$ such that $f|_{B}=\operatorname{id}_{B}$ and, respectively, $f|_{B}=\pi_{B}^{A}\colon B\to A$. By inspection of 2.6 the following triangle commutes

[TABLE]

Lemma 8.2**.**

Let $k\geq 0$. With the notation above, if $\dim S(U)^{H}\geq k+2$ for every $H\in\operatorname{Iso}_{G}(S(U))$ then the diagonal arrow in (8.2) is a $k$-equivalence.

Proof.

By Proposition 2.7(a) $(\eta_{S(U)})^{H}=\eta_{S(U^{H})}$ is a $(2\dim S(U)^{H}-2)$-equivalence for any $H\in\operatorname{Iso}_{G}(S(U))$. The result follows by applying Lemma 2.4 with $\emptyset\subseteq S(U)$ since $2\dim S(U)^{H}-2\geq\dim S(U)^{H}+k$. ∎

Proof of Proposition 1.2.

Combine Lemma 8.2 with parts (1) and (2) of Lemma 8.1. ∎

There is a continuous function

[TABLE]

It gives rise to a homeomorphism $S^{U}\wedge S^{V}\cong S^{U+V}$. Moreover, $(-)\wedge\alpha_{0}$ carries $B\subseteq S^{U}$ to $B\subseteq S^{U+V}$.

Under the homeomorphism $\Sigma S(U)\cong S^{U}$ the map $\operatorname{susp}$ in the square below has the form $\operatorname{susp}(f)(u)=|u|\cdot f(u/|u|)$ and $J_{S(V)}(f)(u+v)=|u|\cdot f(u/|u|)+v$ and $(f\wedge S^{V})(u+v)=f(u)+v$. The square commutes by inspection.

[TABLE]

Proof of Theorem 1.3.

Given $n\geq 1$ we will write $U=U_{\leq n}$ and $V=U_{n+1}$. Lemma 8.2 and Theorem 1.1 imply that if $n$ is large enough then the arrows $\operatorname{susp}$ and $J_{S(V)}$ in (8.3) are $k$-equivalences, and therefore the arrow $(-)\wedge S^{V}$ on the right is a $k$-equivalence.

The inclusions $A\subseteq B\subseteq S^{U}$ give rise to the following fibrations, with the inclusion maps as basepoints,

[TABLE]

and similar ones for $S^{U+V}$. The fibres of fibrations \textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}} with basepoints $x_{0},x_{1},x_{2}$ fit into a fibre sequence \textstyle{F_{12}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{02}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{01}} . We obtain a commutative diagram

[TABLE]

Assume first that $\operatorname{Irr}(U_{\bullet})$ contains the trivial representation. By Lemma 8.1, if $n$ is large enough then $G\in\mathcal{F}(U_{\bullet})=\operatorname{Iso}_{G}(U)$ and $\dim(S^{U})^{G}\geq k+2$. Therefore the inclusion of the fibres in the square above are $k$-equivalences, hence $(-)\wedge S^{V}$ in the 2nd column are ones too. Together with Theorem 1.1 and since $\operatorname{susp}$ in (8.3) is a $k$-equivalence, the result follows.

If $\operatorname{Irr}(U_{\bullet})$ does not contain the trivial representation then $G\notin\mathcal{F}(U_{\bullet})$ and therefore $(S^{U})^{G}\cong(S^{U+V})^{G}=\{\alpha_{0},\alpha_{1}\}$. Then $\operatorname{map}^{G}_{*}(S^{U},S^{U})$ is the disjoint union of the fibres of $\operatorname{ev}_{\alpha_{0}}$ over $\alpha_{0}$ and over $\alpha_{1}$, i.e $\operatorname{map}^{G}(S^{U},S^{U};\operatorname{id}_{B})$ and $\operatorname{map}^{G}(S^{U},S^{U};\pi_{B}^{A})$ respectively. Similarly $\operatorname{map}^{G}_{*}(S^{U+V},S^{U+V})$ has such decomposition. We have seen that $(-)\wedge S^{V}$ induces a $k$-equivalence on the components over $\operatorname{id}_{B}$. It induces a $k$-equivalence on the components over $\pi_{B}^{A}$ by Lemma 8.4 below. The result follows from Theorem 1.1 and since $\operatorname{susp}$ in (8.3) is a $k$-equivalence. ∎

Consider $G$-spaces $(X;x_{0},x_{1})$ with distinguished points $x_{0},x_{1}\in X^{G}$. Examples are given by $(\Sigma X;\alpha_{0},\alpha_{1})$, see 2.6. Given $(X;x_{0},x_{1})$ and $(Y;y_{0},y_{1})$ let

[TABLE]

Let $[t,x]$ denote the equivalence classes of the points of $\Sigma X$. There is a pinch map $\Sigma X\to\Sigma X\curlyvee\Sigma X$ where $[t,x]\mapsto[2t,x]$ if $0\leq t\leq\frac{1}{2}$ and $[t,x]\mapsto[2t-1,x]$ if $\frac{1}{2}\leq t\leq 1$. For any $G$-space $Z$ there results

[TABLE]

The space on the left is identified with the space of pairs $(f,g)$ of maps $\Sigma X\to Z$ such that $f(\alpha_{1})=g(\alpha_{0})$. We will denote $\operatorname{pinch}^{*}(f,g)$ by $f+g$.

Let $\operatorname{inv}\colon\Sigma X\to\Sigma X$ be the map $[t,x]\mapsto[1-t,x]$. Let $A\subseteq B\subseteq\Sigma X$ denote $\{\alpha_{1}\}$ and $\{\alpha_{0}\,\alpha_{1}\}$ respectively. Let $\operatorname{map}^{G}(\Sigma X,\Sigma X;\operatorname{id}_{B})$ be the space of maps with $f|_{B}=\operatorname{id}_{B}$. Similarly $\operatorname{map}^{G}(\Sigma X,\Sigma X;\pi_{B}^{A})$ is the space of maps with $f|_{B}=\pi_{B}^{A}\colon B\to A$. Define maps

[TABLE]

Lemma 8.3**.**

The maps $\tau$ and $\sigma$ in (8.4) are equivariant homotopy equivalences.

Proof.

First, $(\operatorname{id}+\operatorname{inv})\colon\Sigma X\to\Sigma X$ is equivariantly homotopic to the constant map $\alpha_{0}$ (via the homotopy $h_{s}([t,x])=[2ts,x]$ if $0\leq t\leq\frac{1}{2}$ and $h_{s}([t,x])=[s(2-2t),x]$ if $\frac{1}{2}\leq t\leq 1$). Similarly $(\operatorname{inv}+\operatorname{id})$ is homotopic to the constant map $\alpha_{1}$.

Given $f\colon\Sigma X\to\Sigma X$ such that $f(\alpha_{0})=\alpha_{0}$, there is a natural homotopy $\alpha_{0}+f\simeq f$. Similarly, if $f(\alpha_{0})=\alpha_{1}$ there is a natural homotopy $\alpha_{1}+f\simeq f$.

There is a natural homotopy between the maps

[TABLE]

Thus, there are homotopies $\tau(\sigma(f))=\operatorname{id}+(\operatorname{inv}+f)\simeq(\operatorname{id}+\operatorname{inv})+f\simeq\alpha_{0}+f\simeq f$ and $\sigma(\tau(f))=\operatorname{inv}+(\operatorname{id}+f)\simeq(\operatorname{inv}+\operatorname{id})+f\simeq\alpha_{1}+f\simeq f$ natural in $f$. ∎

Neighbourhoods of $\infty\in S^{U}$ are the open balls $\{u\in U:|u|>R\}\cup\{\infty\}$.

Lemma 8.4**.**

Let $U,V$ be (orthogonal) representations of $G$. There is a homotopy commutative square in which the horizontal maps are homotopy equivalences

[TABLE]

Proof.

Since $\Sigma S(U)\cong S^{U}$ the maps $\tau$ in (8.4) are homotopy equivalences by Lemma 8.3.

Fix once and for all a homeomorphism $\varphi\colon[0,1]\to[0,\infty]$ with $\varphi(0)=0$. Let $U\oplus V$ be the orthogonal sum (thus, $|u+v|=|u|+|v|$). We model the pinch map $S^{U}\to S^{U}\curlyvee S^{U}$ by

[TABLE]

Define $h\colon I\times\operatorname{map}^{G}(S^{U},S^{U};\pi_{B}^{A})\times S^{U+V}\to S^{U+V}$ where $A=\{\infty\}$ and $B=\{0,\infty\}\subseteq S^{U}$:

[TABLE]

This is well defined because if $|u+tv|=1$ then $\infty\cdot(u+tv)=\infty$ and $f(0)=\pi_{B}^{A}(0)=\infty$. In what follows we will show that $h$ is continuous. Once this is established, the adjoint of $h$ gives a map $H\colon I\times\operatorname{map}^{G}(S^{U},S^{U};\pi_{B}^{A})\to\operatorname{map}^{G}(S^{U+V},S^{U+V};\operatorname{id}_{B})$ since $h(t,f,0)=0$ and $h(t,f,\infty)=\infty$. It is a homotopy from $(-)\wedge S^{V}\circ\tau$ to $\tau\circ(-)\wedge S^{V}$ in the square above, which completes the proof. To show $h$ is continuous we apply the pasting lemma to the following subsets of $I\times\operatorname{map}^{G}(S^{U},S^{U};\pi_{B}^{A})\times S^{U+V}$

[TABLE]

Claim 1: $D$ and $E$ are is a closed subsets of $I\times\operatorname{map}^{G}(S^{U},S^{U};\pi_{B}^{A})\times S^{U+V}$.

Proof: We replace $D$ and $E$ with their images in $I\times S^{U+V}$ under the projection. The complement of $E$ is $\{(t,u+v):u+v\neq\infty\text{ and }|u+tv|<1\}$ which is clearly open. The complement of $D$ is the preimage of $(1,\infty]$ under the map $\lambda\colon I\times S^{U+V}\setminus\{(0,\infty)\}\xrightarrow{(t,u+v)\mapsto|u+tv|}[0,\infty]$ where $\lambda(t,\infty)=\infty$. It is clearly continuous at any $(t_{0},u_{0}+v_{0})$ with $u_{0}+v_{0}\neq\infty$; It is continuous at points $(t_{0},\infty)$ with $t_{0}>0$ since given $R>0$, whenever $(t,u+v)\in I\times S^{U+V}\setminus\{(0,\infty)\}$ is such that $t>\frac{t_{0}}{2}$ and $|u+v|>R+\frac{2R}{t_{0}}$ we have either $|u|>R$ in which case $|\lambda(t,u+v)|\geq|u|>R$ or $|v|>\frac{2R}{t_{0}}$ so $|\lambda(t,u+v)|\geq t|v|>R$.

Claim 2: $h$ is continuous on $D$.

Proof: We may replace $D$ with its projection $D^{\prime}$ in $I\times S^{U+V}$ and prove that $\lambda\colon D^{\prime}\to S^{U+V}$ defined by $\lambda(t,u+v)=\varphi(|u+tv|)u+(t\varphi(|u+tv|)+1-t)v$ if $u+v\neq\infty$ and $\lambda(0,\infty)=\infty$, is continuous. Continuity is clear away from $(0,\infty)\in D^{\prime}$. Continuity at $(0,\infty)$ will follow once we show that given $R>1$, if $(t,u+v)\in D^{\prime}$ is such that $|u+v|>R+2$ then $|\lambda(t,u+v)|>R$. Indeed, $|u+tv|\leq 1$ implies $|u|,|tv|\leq 1$, hence $|v|>R+1$ and $t\leq\frac{1}{|v|}$, and it follows that $|\lambda(t,u+v)|\geq(t\varphi(|u+tv|)+1-t)|v|\geq(1-t)|v|\geq|v|-1>R$.

Claim 3: $h$ is continuous on $E$.

Proof: Let $E^{\prime}$ be the projection of $E$ in $I\times S^{U+V}$. Define $\lambda\colon E^{\prime}\to S^{U+V}$ by $\lambda(t,u+v)(1-\frac{1}{|u+tv|})u+(1-\frac{t}{|u+tv|})v$ if $u+v\neq\infty$ and $\lambda(t,\infty)=\infty$. Then $h|_{E}(t,f,u+v)=(f\wedge S^{V})(\lambda(t,u+v))$, and since $f\mapsto f\wedge S^{V}$ and the evaluation map are continuous, it remains to show that $\lambda$ is continuous. This is clear away from the points $(t,\infty)$. Continuity of $\lambda$ at points $(t_{0},\infty)\in E^{\prime}$ would follow once we show that given $R>1$, if $(t,u+v)\in E^{\prime}$ is such that $|u+v|>R+2$, then $|\lambda(t,u+v)|>R$. Indeed, $|u+tv|\geq|u|,t|v|$ and therefore $(1-\frac{1}{|u+tv|})|u|\geq|u|-1$ and $(1-\frac{t}{|u+tv|})|v|\geq|v|-1$ (take special care when $u=0$ or $v=0$ or $t=0$). It follows that if $|u+v|>R+2$ then $|\lambda(t,u+v)|\geq|u|-1+|v|-1>R$. ∎