Fixed point sets of equivariant fiber-preserving maps

TL;DR

This paper extends fixed point set characterization results to equivariant fiber-preserving maps on G-spaces, broadening the understanding of fixed points in symmetric topological structures.

Contribution

It proves an equivariant analogue of Brown-Soderlund's theorem for fixed point sets in the category of G-spaces and G-maps, generalizing previous results.

Findings

Established necessary and sufficient conditions for fixed point sets of G-equivariant fiber-preserving maps.

Extended classical fixed point theorems to the setting of G-spaces with group actions.

Provided a framework for analyzing fixed points in symmetric fiber bundle contexts.

Abstract

Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C. Soderlund extended Schirmer's result to the category of fiber bundles and fiber-preserving maps. The objective of this paper is to prove an equivariant analogue of Brown-Soderlund theorem result in the category of $G$-spaces and $G$-maps where $G$ is a finite group.

Equations

$$H(t,\cos \theta \sin \psi ,\sin \theta \sin \psi ,\cos \psi )=(\cos (\theta +tϵ\sin \theta )\sin \psi ,\sin (\theta +tϵ\sin \theta )\sin \psi ,\cos \psi ),$$

$$St(A,K)=\bigcup _{\stackrel{\overline{|t|}\cap A\ne \varnothing }{t\in K}}|t|,$$

$$\{{\overline{H}}^K(p(t),t)\}\sim \underset{=\{f\circ p(t)\}}{\underbrace{\{{\overline{H}}^K(p(t),0)\}}}\ast \{{\overline{H}}^K(p(1),t)\}.$$

$${\overline{H}}_3:(X\times \{0\})\cup (St(A)\times I)\to X$$

$$\begin{array}[]{ccc}(x,t)&\mapsto&\left\{\begin{array}[]{ll}\alpha(x,h_{2}(x),1-(1-\overline{d}(x,\Phi))t)&if\ (x,t)\in St(A)\times I;\\ h_{2}(x)&if\ t=0.\end{array}\right.\end{array}$$

$$\begin{array}[]{c}H\Big{(}\big{(}\cos\theta\sin\psi{\bf,}\sin\theta\sin\psi,\cos\psi\big{)},\cos x+i\sin x,t\Big{)}=\\ \Big{(}\big{(}-\cos(\theta+t|\sin x|\pi)\sin\psi,-\sin(\theta+t|\sin x|\pi)\sin\psi,-\cos\psi\big{)},\cos x+i\sin x\Big{)}.\end{array}$$

$$\begin{array}[]{ccclccl}H^{\prime}:&X&\rightarrow&E^{I}&&&\\ &x&\mapsto&H^{\prime}(x,\bullet):&I&\rightarrow&E\\ &&&&t&\mapsto&H^{\prime}(x,t).\end{array}$$

$$\begin{array}[]{rcl}\overline{h}_{t}(x)&=&\left\{\begin{array}[]{ll}g\overline{H}_{i}(g^{-1}x,t),&for\ x\in X-A,\ where\ G_{x}=gWK_{i}g^{-1};\\ \overline{h}(x),&for\ x\in V.\end{array}\right.\end{array}$$

$${\rm Fix}(\overline{H}_{i}(\bullet,1))=p(A)\cup\big{(}\bigcup_{j\in\ T,\ j\leq\ i}(G\{b_{j,1}\}\cup\dots\cup G\{b_{j,m_{j}}\})\ \big{)}$$

$$\{\overline{\alpha }\}\sim \{{\overline{f}}^K\circ \overline{\alpha }\}\ast \{{\overline{H}}_A^K(\overline{\alpha }(1),t)\}\sim \{{\overline{H}}^K(\overline{\alpha }(t),t)\}.$$

$$H^{\prime }(x,0)=\overline{H}(p(x),0)=\overline{f}\circ p(x)=p\circ f(x).$$

$$\mathrm{Fix}(h_1)\subset p^{-1}(\mathrm{Fix}(\overline{h}))=p^{-1}(p(A)\cup G\{b_1\}\cup \cdots \cup G\{b_l\}).$$

$${\tilde{H}}_2:(X\times \{0\})\cup (p^{-1}(\mathrm{Fix}(\overline{h}))\times I)\to X$$

$$\begin{array}[]{lcr}\widetilde{H}_{2}(x,t)&=&\left\{\begin{array}[]{ll}h_{1}(x)&if\ t=0\ or\ if\ x\in p^{-1}(p(A));\\ H_{2,b_{j}}(x,t)&if\ x\in p^{-1}(G\{b_{j}\}).\end{array}\right.\end{array}$$

$$\begin{array}[]{ccl}\widetilde{H}(x,t)&=&\left\{\begin{array}[]{ll}h_{2}(x),&if\ t=0;\\ h_{2,b_{j}}(x),&if\ x\in Gp^{-1}(b_{j});\\ \widetilde{H}_{A}(x,t),&if\ x\in A.\end{array}\right.\end{array}$$

$$\pi \circ f^{\ast }=\underset{=p}{\underbrace{(p\circ \Phi )\circ ({\Phi }^{-1}}}\circ f\circ \Phi )=\underset{=p}{\underbrace{p\circ f}}\circ \Phi =p\circ \Phi =\pi .$$

$$U^{*}(b,y,t)=\left\{\begin{array}[]{lc}(b,f^{*}_{D(b,2t)}(y)),&if\ 0\leq t\leq\frac{1}{2}\\ (b,f^{*}_{D(b_{0},2-2t)}(y)),&if\ \frac{1}{2}\leq t\leq 1.\\ \end{array}\right.$$

$$f^{\ast }(b,y)={\Phi }^{-1}\circ \underset{=\Phi (b,y)}{\underbrace{f\circ \underset{\in A}{\underbrace{\Phi (b,y)}}}}={\Phi }^{-1}\circ \Phi (b,y)=(b,y).$$

$$({\Phi }^{-1}{)}^K(A^K\cap WK(p^K{)}^{-1}(b_0))=WK\{b_0\}\times Y_0^K.$$

$$H^{*}(b,y,t)=\left\{\begin{array}[]{lc}U^{*}(b,y,2t),&if\ 0\leq t\leq\frac{1}{2}\\ (b,V^{*}(y,2t-1)),&if\ \frac{1}{2}\leq t\leq 1.\\ \end{array}\right.$$

$$\begin{array}[]{ccl}\widetilde{H}_{2}(x,t)&=&\left\{\begin{array}[]{ll}f(x),&if\ t=0\ or\ p(x)\in F\\ H_{j}(x,t),&if\ p(x)\in p(A)_{j}.\end{array}\right.\end{array}$$

Full text

keywords: fixed points, equivariant maps

AMS subj. class. [2010]: Primary: 55M20; Secondary: 57S99

1 Introduction and statement of results

A well-known and important question in classical topology is the fixed point property. Recall that a topological space $X$ is said to have the fixed point property if every (continuous) map $f:X\to X$ must have a fixed point $x_{0}\in X$ such that $f(x_{0})=x_{0}$. A related question is the so-called complete invariance property for deformation (CIPD). We say that $X$ has the CIPD if for any nonempty closed subset $A\subset X$, there exists a selfmap $f:X\to X$ homotopic to the identity $1_{X}$ such that $A={\rm Fix}(f)=\{x\in X\mid f(x)=x\}$. In [9], H. Schirmer generalized the concept of CIPD and gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map $g$ in the homotopy class of a given selfmap $f$. That is, given a map $f:X\to X$, Schirmer determined when a closed nonempty subset $A$ can be realized as $A={\rm Fix}(g)$ for some $g$ homotopic to $f$. Upon relaxing the conditions given by Schirmer, C. Soderlund together with R. Brown [3] generalized Schirmer’s result to fiber-preserving maps of fiber bundles.

Suppose that $X$ is a compact connected polyhedron without local cutpoints and $A$ is a closed subset imbedded inside a subpolyhedron $K$ that can be by-passed in $X$, that is, every path $C$ in $X$ with $C(0),C(1)\in X-K$, is homotopic to a path $C^{\prime}$ in $X-K$ relative to the endpoints. H. Schirmer [9] introduced the following two conditions to realize $A$ as the fixed point set of a selfmap in the homotopy class $[f]$.

1. (C1)

if there exists a homotopy $H_{A}:A\times[0,1]\to X$ from $f|_{A}$ to the inclusion $i:A\hookrightarrow X$; 2. (C2)

if for every essential fixed point class $\mathbb{F}$ of $f$, there exists a path $\alpha:[0,1]\to X$ with $\alpha(0)\in\mathbb{F},\alpha(1)\in A$ and $\{\alpha(t)\}\sim\{f\circ\alpha(t)\}\ast\{H_{A}(\alpha(1),t)\}$ relative to the endpoints.

Soderlund [10, Theorem 3.5] showed, by relaxing the assumption on $A$ given by Schirmer, the following result.

Theorem 1.1**.**

Let $X$ be a compact, connected polyhedron with no local cut points and $A$ be a closed locally contractible subspace of $X$ such that $X-A$ is not a $2$-manifold and $A$ can be by-passed in $X$. Then $A={\rm Fix}(g)$ for some $g\sim f$ if and only if (C1) and (C2) are satisfied.

Subsequently, R. Brown and C. Soderlund [3] introduced analogous conditions in the fiber-preserving setting. Let $\mathfrak{F}=(E,p,B;Y)$ be a (locally trivial) fiber bundle and $f:E\to E$ a fiber preserving map.

1. (${\rm C1}_{\mathfrak{F}}$)

if there exists a fiber preserving homotopy $H_{A}:A\times[0,1]\to E$ from $f|_{A}$ to the inclusion $i:A\hookrightarrow E$; 2. (${\rm C2}_{\mathfrak{F}}$)

if for every essential fixed point class $\mathbb{F}$ of $f$, there exists a path $\alpha:[0,1]\to E$ with $\alpha(0)\in\mathbb{F},\alpha(1)\in A$ and $\{\alpha(t)\}\sim\{f\circ\alpha(t)\}\ast\{H_{A}(\alpha(1),t)\}$ relative to the endpoints.

Following the terminology of [3], we call $(X,A)$ a suitable pair if $X$ is a finite polyhedron with no local cut points and $A$ is a closed locally contractible subspace of $X$ such that $X-A$ is not a $2$-manifold and $A$ can be by-passed in $X$.

In [3], it was shown that conditions $({\rm C1}_{\mathfrak{F}})$ and $({\rm C2}_{\mathfrak{F}})$ are also sufficient. The following is their main result.

Theorem 1.2**.**

Let $\mathfrak{F}=(E,p,B;Y)$ be a fiber bundle where $E,B$ and $Y$ are connected finite polyhedra, $f:E\to E$ a fiber preserving map and $A$ a closed locally contractible sub-bundle of $E$ such that each component $p(A)_{j}$ of $p(A)$ is contractible and $(B,p(A)),(Y,Y_{j})$ for all sub-bundle fibers $Y_{j}$ of $A$, are suitable pairs. Suppose $({\rm C1}_{\mathfrak{F}})$ and $({\rm C2}_{\mathfrak{F}})$ are satisfied and $A$ intersects every essential fixed point class of $f_{b_{j}}:p^{-1}(b_{j})\to p^{-1}(b_{j})$ for at least one $b_{j}$ in each component $p(A)_{j}$. If $Z$ is a closed bundle subset of $A$ that intersects every component of $A$, then there exists a map $g:E\to E$ that is fiber preserving and fiberwise homotopic to $f$ ($g\sim_{\mathfrak{F}}f$) such that ${\rm Fix}(g)=Z$.

In particular, when $Z=A$, this theorem shows that $({\rm C1}_{\mathfrak{F}})$ and $({\rm C2}_{\mathfrak{F}})$ are necessary and sufficient for $A={\rm Fix}(g)$ for some $g\sim_{\mathfrak{F}}f$.

Many applications involve symmetries in the presence of a group action. As a result, equivariant topology has been proven to be useful in the study of nonlinear problems. In the equivariant setting, we are concerned with a group $G$ acting on a space $X$ together with a $G$-map $f:X\rightarrow X$ which respects the group action, that is, for all $\alpha\in G$, $f(\alpha x)=\alpha f(x)$ for all $x\in X$. In this case, the fixed point set ${\rm Fix}(f)$ is a priori a $G$-invariant subset of $X$.

In [9], Schirmer observed that for a given selfmap $f:\mathbb{S}^{n}\rightarrow\mathbb{S}^{n}$ of an $n$-sphere, $n\geq 2$, any closed nonempty proper subset $A$ of $\mathbb{S}^{n}$ can be realized as the fixed point set of a map $g\in[f]$ with ${\rm Fix}(g)=A$. However, such phenomenon does not hold if we impose a group action as we show in the following example, which gives the underlying motivation for this paper.

Example 1.3**.**

Let $G=\mathbb{Z}_{2}$, $X=\mathbb{S}^{2}$ and the action is given by $\xi(x,y,z)\mapsto(-x,-y,z)$. If $A=\{(x,y,0)\in\mathbb{S}^{2}\}$ then $A$ is $\mathbb{Z}_{2}$-invariant, but there is no $\mathbb{Z}_{2}$-map $h:\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$ that is $\mathbb{Z}_{2}$-homotopic to the identity map $Id:\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$ such that ${\rm Fix}(h)=A$.

In fact, suppose there is a $\mathbb{Z}_{2}$-homotopy $H$ from $Id$ to $h:\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$ such that ${\rm Fix}(h)=A$. Then, $h$ preserves $X^{G}=\{N,S\}$, where $N=(0,0,1)$ and $S=(0,0,-1)$. Hence, $h(N)=S$ and $h(S)=N$ and the path $p:I\rightarrow X^{G}$ defined by $p(t)=H^{G}(N,t)$ is such that $p(0)=Id(N)=N$ and $p(1)=h(N)=S$. But, this is impossible.

In this situation, the location of $A$ in $X$ is more important than its topology, because if we replace $A$ by $A^{\prime}=\{(x,0,z)\in\mathbb{S}^{2}\}$ then:

[TABLE]

is a $\mathbb{Z}_{2}$-homotopy (with polar coordinates) between the identity and the $\mathbb{Z}_{2}$-map $h$ such that ${\rm Fix}(h)=A^{\prime}$.

Example 1.4**.**

It is easy to see, by modifying the last example, that the equivariant analogue of Schirmer’s result does not hold in general. Let $G=\mathbb{Z}_{2}$, $X=\mathbb{S}^{2}\times\mathbb{S}^{2}$ and the action is given by $\xi((x,y,z),(x^{\prime},y^{\prime},z^{\prime}))\mapsto((-x,-y,z),(1,0,0))$. The set $A=\{((x,y,0),(1,0,0))\in X\}$ is $\mathbb{Z}_{2}$-invariant and $X^{G}=\{N,S\}\times\{(1,0,0)\}$ consists of two points. The same argument as in Example 1.3 shows that $A$ cannot be the fixed point set of any map $\mathbb{Z}_{2}$-homotopic to the identity map while $(X,A)$ satisfies the conditions of Schirmer’s result for $A$ can be by-passed since $A$ has codimension $3$ in $X$.

The main objective of this paper is to give an equivariant analogue of Schirmer’s result and of Brown-Soderlund’s result. This paper is organized as follows. In the first section, we briefly recall the non-equivariant results of [9] and [3] and review some basic background on $G$-maps and $G$-spaces where $G$ denotes a compact Lie group. Then we review the necessary equivariant Nielsen fixed point theory from [13]. In section 2, we prove our first main result, an equivariant analogue of [9]:

Theorem 1.5**.**

Let $G$ be a compact Lie group, $X$ be a compact and smooth $G$-manifold and $A$ be a nonempty, closed, locally contractible $G$-subset of $X$ such that for each finite $WK$ we assume that $\dim(X^{K})\geq 3$, $\dim(X^{K})-\dim(X^{K}-X_{K})\geq 2$ and $A^{K}$ is by-passed in $X^{K}$, for all $(K)\in{\rm Iso}(X)$. Suppose that the following conditions holds for a $G$-map $f:X\rightarrow X$:

• ($C_{G}1$)

there exists a $G$-homotopy $H_{A}:A\times I\rightarrow X$ from $f|_{A}$ to the inclusion $i:A\hookrightarrow X$;

• ($C_{G}2$)

for each finite $WK$, for every $WK$-essential fixed point class $F$ of $f^{K}:X^{K}\rightarrow X^{K}$ there exists a path $\alpha:I\rightarrow X^{K}$ with $\alpha(0)\in F$, $\alpha(1)\in A^{K}$, and $\{\alpha(t)\}\sim\{f^{K}\circ\alpha(t)\}\ast\{H_{A}^{K}(\alpha(1),t)\}.$

Then for every closed $G$-subset $\Phi$ of $A$ that has nonempty intersection with every component of $A$ there exists a $G$-map $h:X\rightarrow X$, $G$-homotopic to $f$ with ${\rm Fix}(h)=\Phi$.

In the last section, we apply Theorem 1.5 to prove an equivariant analogue of [3] when $G$ is finite:

Theorem 1.6**.**

Let $G$ be a finite group, $\mathfrak{F}=(X,p,B,Y)$ be a $G$-fiber bundle where $X$, $B$ and $Y$ are compact and smooth $G$-manifolds, $\dim(B^{K})\geq 3$, $\dim(B^{K})-\dim(B^{K}-B_{K})\geq 2$, for all $(K)\in{\rm Iso}(B)$, $\dim(Y^{K})\geq 3$, $\dim(Y^{K})-\dim(Y^{K}-Y_{K})\geq 2$, for all $(K)\in{\rm Iso}(Y)$.

Let $A$ be a nonempty, closed, locally contractible $G$-subset of $X$ such that $(X,A)$ is $G$-fiber bundle pair with respect to the fiber bundle $\mathfrak{F}$, $p(A)$ be a closed $G$-subset of $B$ such that each component $p(A)_{j}$ of $p(A)$ is equivariantly contractible and $p^{K}(A^{K})$ is by-passed in $B^{K}$, for all $(K)\in{\rm Iso}(B)$. Let $Y_{j}$ be a subbundle fiber of $A$ such that $Y_{j}$ is a closed and locally contractible $G$-subset of $Y$ and $Y^{K}_{j}$ is by-passed in $Y^{K}$, for all $(K)\in{\rm Iso}(Y)$, and $f:X\rightarrow X$ be a $G$-fiber-preserving map such that $A^{K}$ intersects every essential $WK$-fixed point class of $f^{K}_{b_{j}}:WK(p^{K})^{-1}(\{b_{j}\})\rightarrow WK(p^{K})^{-1}(\{b_{j}\})$ for at least one $b_{j}$ in each component $p^{K}(A^{K})_{j}$, for all $(K)\in{\rm Iso}(X)$. Suppose that the following conditions hold for $f$ and $A$:

• $(C_{G}1)_{\mathfrak{F}}$

there exists a $G$-fiberwise-homotopy $H_{A}:A\times I\rightarrow X$ from $f|_{A}$ to the inclusion $i:A\hookrightarrow X$;

• $(C_{G}2)_{\mathfrak{F}}$

for every $WK$-essential fixed point class $F$ of $f^{K}:X^{K}\rightarrow X^{K}$ there exists a path $\alpha:I\rightarrow X^{K}$ with $\alpha(0)\in F$, $\alpha(1)\in A^{K}$, and $\{\alpha(t)\}\sim\{f^{K}\circ\alpha(t)\}\ast\{H_{A}^{K}(\alpha(1),t)\}.$

Then for every nonempty closed $G$-bundle subset $\Phi$ of $A$ that intersects every component of $A$ there exists a $G$-fiber-preserving map $h$, $G$-fiberwise homotopic to $f$ with ${\rm Fix}(h)=\Phi$.

In order to establish the notations, let $G$ be a topological group and $X$ be a (left) $G$-space. Given a subgroup $K$ of $G$ we denote by $NK$ the normalizer of $K$ in $G$, $WK=\frac{NK}{K}$ is the Weyl group of $K$ in $G$. The orbit type of $K$ is the conjugacy class of $K$ in $G$ denoted by $(K)$. If $(K_{1})$ is subconjugate to $(K_{2})$, we write $(K_{1})\leq(K_{2})$.

If $x\in X$, then $G_{x}=\{g\in G;\ gx=x\}$ denotes the isotropy subgroup of $x\in X$, and $(G_{x})$ is called an isotropy type of $X$. We denote by ${\rm Iso}(X)$ the set of isotropy types of $X$. Moreover, $X^{K}=\{x\in X;K\leq G_{x}\}$, $X^{(K)}=\{x\in X;(K)\leq(G_{x})\}$, $X_{K}=\{x\in X;G_{x}=K\}$ and $X_{(K)}=\{x\in X;G_{x}\subset(K)\}$.

If ${\rm Iso}(X)$ is finite (in particular when $G$ is finite), we can choose an admissible ordering on ${\rm Iso}(X)$ such that $(K_{i})\leq(K_{j})$ implies $i\leq j$. Then we have a filtration of $G$-subspaces $X_{1}\subset\dots\subset X_{n}=X$ where $X_{i}=\{x\in X;\ (G_{x})=(H_{j})$ for some $j\leq i\}$

If $f:X\to X$ is a $G$-map, then $f^{K}=f|_{X^{K}}:X^{K}\to X^{K}$ is a $WK$-map. Let $\mathcal{F}=\{(K)\in{\rm Iso}(X)\mid|WK|<\infty\}$ and $(K)\in\mathcal{F}$. If $x,y\in{\rm Fix}(f^{K})$ then $x\sim_{K}y$ if either $y=\alpha x$ for some $\alpha\in WK$ or $\exists\ \sigma:[0,1]\to X^{K}$ such that $\sigma\sim f^{K}\circ\sigma$ relative to endpoints. Then $\sim_{K}$ is an equivalence relation on ${\rm Fix}(f^{K})$ and the equivalence classes are called the $WK-$fixed point classes of $f^{K}$. Evidently, a $WK$-fixed point class $\mathcal{W}$ is a disjoint union of a finite number of ordinary fixed point classes $W_{1},...,W_{r}$ of $f^{K}$ and thus the fixed point index $ind(\mathcal{W})$ is defined as $ind(\mathcal{W})=\sum_{i}ind(W_{i})$. A $WK$-fpc (fixed point class) $\mathcal{W}$ is essential if $ind(\mathcal{W})\neq 0$. For further information on equivariant Nielsen fixed point theory, see [13]. Throughout, by a smooth $G$-manifold $X$, we assume that the fixed point set $X^{H}$ is a smooth connected submanifold for each isotropy subgroup $H\leq G$.

2 Proof of Theorem 1.5 - An equivariant analogue of a result of Soderlund-Schirmer

If $X$ is a smooth $G$-manifold and $A$ is a closed smooth $G$-submanifold of $X$, $G$ being a finite group, then there exists a smooth equivariant triangulation $f_{1}:(K,K_{0})\rightarrow(X,A)$ as proved in [8]. If $B$ is another closed smooth $G$-submanifold of $X$ then there is a smooth equivariant triangulation $f_{2}:(L,L_{0})\rightarrow(X,B)$ and $G$-subdivisions $K^{\prime}$ of $K$ and $L^{\prime}$ of $L$ such that $f^{\prime-1}_{1}\circ f^{\prime}_{2}:|L^{\prime}|\rightarrow|K^{\prime}|$ is a simplicial $G$-homeomorphism, where $f^{\prime}_{1}$ and $f^{\prime}_{2}$ are smooth $G$-triangulations (see [8]).

By Corollary 3.3.5 of [11] and $G$ being finite, we can find unique $G$-subcomplexes $L^{\prime}_{0}$ of $L$ and $K^{\prime}_{0}$ of $K$ such that $L^{\prime}_{0}$ is a refinement of $L_{0}$ and $K^{\prime}_{0}$ is a refinement of $K_{0}$. Then, $f^{\prime-1}_{1}\circ f^{\prime}_{2}(L^{\prime}_{0})=K_{1}$ is a $G$-subcomplex of $K^{\prime}$ and a $G$-triangulation of $B$. In fact, $f^{\prime}_{1}(K_{1})=f^{\prime}_{1}\circ f^{\prime-1}_{1}\circ f^{\prime}_{2}(L^{\prime}_{0})=f^{\prime}_{2}(L^{\prime}_{0})=B$. Hence, by induction if $\{A_{i}\}_{i=1}^{n}$ is a finite collection of closed smooth $G$-submanifolds of $X$ then there exists a smooth equivariant triangulation $f:K\rightarrow X$ and a finite collection of $G$-subcomplexes $\{L_{i}\}_{i=1}^{n}$ of $K$ such that $L_{i}$ is a $G$-triangulation of $A_{i}$, for $i=1,\dots,n$.

To realize $A$ as the fixed point set of some $h:X\rightarrow X$, it is necessary to remove every fixed point $x\in X$ of $f:X\rightarrow X$ outside of $A$. Hence, we need to extend the notion of neighborhood by-passed for a closed subset $A$ as in [10, Definition 2.1] in order to handle these undesired fixed points.. Thus, a $G$-invariant subset $A$ is said to be $G$-neighborhood by-passed if there exists an invariant open subset $U\subset X$ such that $A\subset U$ and $U$ can be by-passed in $X$.

We observe that if $(X,A)$ is a $G$-ENR pair then $A$ is an invariant neighborhood retract in $X$ and if $\{Y_{i}\}_{i=1}^{n}$ is a finite collection of closed smooth $G$-submanifolds such that $\dim(Y_{i})+1<\dim(X)$, then $A\cup Y$ ($Y=\bigcup_{i=1}^{n}Y_{i}$) remains a by-passed $G$-subset of $X$ provided $A$ is by-passed in $X$. Furthermore, a close inspection of the proof of Theorem 2.2 of [10] indicates that the same argument works for the same result in the equivariant setting. That is, if $A$ is a by-passed locally contractible $G$-subset of $X$ then $A$ is $G$-neighborhood by-passed, for $X$ a compact smooth $G$-manifold with $\dim(X)\geq 3$. To see that, we note that if $K$ is the $G$-triangulation of $X$ then there is a by-passed neighborhood (may not be equivariant) $U$ of $A$ in $|K|$. We obtain the open $G$-subpolyhedron:

[TABLE]

such that $\overline{St(A,K)}$ is a subset of $U$ by taking a $G$-refinement $K^{\prime}$ of $K$ if necessary, where $t$ is a simplex of $K$. Therefore, if $p:I\rightarrow X$ is a path with endpoints in $U-\overline{St(A,X)}$ and outside $\overline{St(A,X^{\prime})}$ then using Corollary 3.3.11 of [11] we deform $p$ out of $\overline{St(A,X^{\prime})}$.

Thus, if $\{Y_{i}\}_{i=1}^{n}$ is a finite collection of closed smooth $G$-submanifolds such that $\dim(Y_{i})+1<\dim(X)$ (thus each $Y_{i}$ has codimension at least $2$ in $X$ so that $Y_{i}$ can be by-passed in $X$), then $A\cup Y$ ($Y=\bigcup_{i=1}^{n}Y_{i}$) remains a by-passed $G$-subset of $X$ using a finite collection of $G$-subcomplex $\{L_{i}\}_{i=1}^{n}$ of $K$ such that $L_{i}$ is a $G$-triangulation of $Y_{i}$, for $i=1,\dots,n$.

The next lemma shows how the fixed points outside $A$ may be removed (see also [7]).

Lemma 2.1**.**

Let $\{Y_{i}\}_{i=1}^{n}$ be a finite collection of closed $G$-submanifolds of the $G$-manifold $X$ such that $\dim(Y_{i})+1<\dim(X)$ and the action of $G$ outside $Y=\bigcup_{i=1}^{n}Y_{i}$ is free, where $G$ is a finite group. Let $f:X\rightarrow X$ be a $G$-selfmap, $A$ be a non-empty closed locally contractible and by-passed $G$-subset of $X$ such that $A\subset{\rm Fix}(f)$, there are no fixed points of $f$ in $Y-A$, and $f$ has a finite number of fixed points in $X-(A\cup Y)$. Let $x_{0}$ and $x_{1}$ be two fixed points of $f$ that are $G$-Nielsen equivalent from different orbits such that $x_{0}\in X-(A\cup Y)$ and $x_{1}\in X-(A\cup Y)$ or $x_{1}\in\partial(A)$, where $\partial(A)$ is the boundary of $A$ in $X$ and $q:I\rightarrow X$ a path with end points $q(0)=x_{0}$ and $q(1)=x_{1}$ such that $f\circ q$ is homotopic to $q$ relative to the endpoints.

Then, $f$ is $G$-homotopic, relative to $(A\cup Y)$, to a $G$-selfmap $h:X\rightarrow X$ such that ${\rm Fix}(h)={\rm Fix}(f)-G\{x_{0}\}$.

**Proof of Lemma 2.1: ** Since $A$ is locally contractible and can be by-passed in $X$, the discussion above shows that $A$ is $G$-neighborhood by-passed in $X$. Furthermore, $A\cup Y$ can be by-passed in $X$. Thus, the path $q$ is homotopic, relative to endpoints, to a path $q^{\prime}(t)$ such that for $0\leq t<1$, $q^{\prime}(t)\in X-(A\cup Y)$ with $q^{\prime}(0)=x_{0},q^{\prime}(1)=x_{1}$. Since $G$ acts freely on $X-Y$ and hence on $X-(A\cup Y)$, taking the $G$-translates of $q^{\prime}$ yields $|G|$ paths from the orbit $G\{x_{0}\}$ to the orbit $G\{x_{1}\}$. Note that the segements $G\{q^{\prime}([0,1))\}$ are disjoint while $\{G\{q^{\prime}(1)\}\}$ consists of $[G:G_{x_{1}}]$ distinct endpoints. Here, the isotropy subgroup $G_{x_{1}}$ at $x_{1}$ is trivial if $x_{1}\in X-(A\cup Y)$. Now we coalesce these two fixed orbits in the same fashion as in [14, Lemma 3.1]. (For slightly more general spaces in which normal arcs are used, see [7, Theorem 2].)

$\Box$

We will prove Theorem 2.2 before Theorem 1.5 and for the same reason we prove Theorem 2.2 by first establishing Lemma 2.3 and Lemma 2.5.

Theorem 2.2**.**

Let $G$ be a compact Lie group, $X$ be a compact smooth $G$-manifold and $A$ be a nonempty, closed, locally contractible $G$-subset of $X$ such that for each finite $WK$ we assume that $\dim(X^{K})\geq 3$, $\dim(X^{K})-\dim(X^{K}-X_{K})\geq 2$ and $A^{K}$ is by-passed in $X^{K}$, for all $(K)\in{\rm Iso}(X)$. Then, given a $G$-map $f:X\rightarrow X$ there exists a $G$-map $h:X\rightarrow X$ $G$-homotopic to $f$ with ${\rm Fix}(h)=A$ if, and only if, the conditions $(C_{G}1)$ and $(C_{G}2)$, given in Theorem 1.5, hold for $f$ relative to $A$.

Lemma 2.3**.**

Let $G$ be a compact Lie group, $X$ be a $G$-space $G$-ANR and $A$ be a nonempty closed $G$-subset of $X$. If $f:X\rightarrow X$ is a $G$-map $G$-homotopic to $h:X\rightarrow X$ such that ${\rm Fix}(h)=A$ then the conditions $(C_{G}1)$ and $(C_{G}2)$ given by Theorem 1.5 hold for $f$ relative to $A$.

Proof of Lemma 2.3: Let $H:X\times I\rightarrow X$ be a $G$-homotopy which starts at $f$ and ends at $h$. Then $\overline{H}=H|_{(X\times\{0\})\cup(A\times I)}:(X\times\{0\})\cup(A\times I)\rightarrow X$ satisfies $(C_{G}1)$. If $F$ is a $WK$-essential fixed point class of $f^{K}$, then, there exists a path $p:I\rightarrow X^{K}$ such that $p(0)\in F$ and $p(1)\in J$, where $J\subset A^{K}$ is a $WK$-essential fixed point class of $h^{K}$, $H^{K}$-related to $F$ and $\{p(t)\}\sim\{\overline{H}^{K}(p(t),t)\}$. In fact,

[TABLE]

So, $(C_{G}2)$ is satisfied.

$\Box$

Lemma 2.3 shows that the conditions $(C_{G}1)$ and $(C_{G}2)$ are necessary for $A={\rm Fix}(h)$. The example below shows that these two conditions are independent of each other.

Example 2.4**.**

Let $G=\mathbb{Z}_{2}$, $X=\mathbb{S}^{2}$ and the action given by $\xi(x,y,z)\mapsto(-x,-y,z)$. Then, there is no $\mathbb{Z}_{2}$-homotopy $H$ from the identity $Id$ to $h$ such that ${\rm Fix}(h)=\{(x,y,0)\in\mathbb{S}^{2}\}$. Note that $(C_{G}1)$ occurs, because the map is the identity, but $(C_{G}2)$ does not. On the other hand, let $G=\mathbb{Z}_{2}$, $X=\mathbb{S}^{3}$ and the action given by $\xi(x,y,z,w)\mapsto(x,y,z,-w)$. Then, there is no $\mathbb{Z}_{2}$-homotopy $H$ from the antipodal map $-Id$ to $h$ such that ${\rm Fix}(h)=\{(x,y,z,0)\in\mathbb{S}^{3}\}$. This time $(C_{G}2)$ holds because the map is fixed point free but $(C_{G}1)$ does not hold.

Lemma 2.5**.**

Let $G$ be a compact Lie group, $X$ be a compact smooth $G$-manifold and $A$ be a nonempty, closed, locally contractible $G$-subset of $X$ such that for each finite $WK$ we assume that $\dim(X^{K})\geq 3$, $\dim(X^{K})-\dim(X^{K}-X_{K})\geq 2$ and $A^{K}$ is by-passed in $X^{K}$, for all $(K)\in{\rm Iso}(X)$. If the conditions $(C_{G}1)$ and $(C_{G}2)$, given in Theorem 1.5, hold for a $G$-map $f:X\rightarrow X$ relative to $A$, then there exists a $G$-map $h:X\rightarrow X$, $G$-homotopic to $f$ with ${\rm Fix}(h)=A$.

Proof of Lemma 2.5: This proof follows the steps of the proof of Theorem 3.2 of [9]. Consider a $G$-map $\overline{H}:(X\times\{0\})\cup(A\times I)\rightarrow X$ given by $(C_{G}1)$. It is possible to extend $\overline{H}$ to a $G$-homotopy {}_{1}\overline{H}_{1}:\Big{(}X\times\{0\}\Big{)}\cup\Big{(}(A\cup X_{1})\times I\Big{)}\rightarrow X. As commented above, there is a closed $G$-invariant neighborhood $V$ of $A_{1}$ inside $X_{1}$ and $V$ retracts onto $A_{1}$ equivariantly. Note that $WK_{1}$ acts freely on $X^{K_{1}}_{1}=X_{K_{1}}$ and ${}_{1}h^{K_{1}}_{1}$ is a $WK_{1}$-map. Hence, if $WK_{1}$ has positive dimension we apply Lemma 3.3 of [12] and Lemma 2.1 of [6] to extend ${}_{1}\overline{H}_{1}$ to a $G$-homotopy $\overline{H}_{1}:(X\times\{0\})\cup((A\cup X_{1})\times I)\rightarrow X$, relative to $V$. Moreover, $h_{1}$ has no fixed points in $X_{1}-A_{1}$ and ${\rm Fix}(h_{1})=A$, where $h_{1}=\overline{H}_{1}(\bullet,1):A\cup X_{1}\rightarrow X$.

On the other hand, if $WK_{1}$ is a finite group then $X^{K_{1}}$ is a $WK_{1}$-polyhedron such that $A_{1}^{K_{1}}$ is a $WK_{1}$-subpolyhedron and $St(A_{1},X^{K_{1}})$ is neighborhood by-passed in $X^{K_{1}}$. We apply Lemma 3.1 of [12] and Lemma 2.1 to obtain a $WK_{1}$-homotopy $H:(A_{1}\cup X_{1})^{K_{1}}\times I\rightarrow X^{K_{1}}$ which can be extended by Lemma 2.1 of [6] to a $G$-homotopy $\overline{H}_{1}:(X\times\{0\})\cup((A\cup X_{1})\times I)\rightarrow X$, relative to $V$, such that $h_{1}$ has no fixed points in $X_{1}-A_{1}$ and ${\rm Fix}(h_{1})=A$, where $h_{1}=\overline{H}_{1}(\bullet,1):A\cup X_{1}\rightarrow X$.

By induction, we may assume that we have a $G$-map $\overline{H}_{i-1}:(X\times\{0\})\cup((A\cup X_{i-1})\times I)\rightarrow X$ such that ${\rm Fix}(h_{i-1})=A$, where $h_{i-1}=\overline{H}_{i-1}(\bullet,1):A\cup X_{i-1}\rightarrow X$ and the proof follows the steps we did for $WK_{1}$.

$\Box$

Now Theorem 2.2 follows easily from Lemma 2.3 and Lemma 2.5.

Proof of Theorem 1.5: First of all, by Theorem 2.2, there is a $G$-map $h_{1}:X\rightarrow X$ $G$- homotopic to $f$ such that ${\rm Fix}(h_{1})=A$. We may apply Proposition 2.5 of [12] and Theorem 4.3 of [13] to conclude that $h_{1}$ is $G$-homotopic to $h_{2}$ such that $h_{2}|_{X^{K}}$ has a finite number of fixed points, all of which inside $St(A^{K})$ and lying in the interior of a maximal simplex of $X^{K}$ and $h_{2}$ is a $G$-proximity map in $St(A)$ (for some $G$-triangulation of $X$).

Since $\Phi$ has nonempty intersection with every component of $A$ we can pull the fixed points of $h_{2}$ to $\Phi$. Let $\alpha$ be the $G$-map of Lemma VIII.C.1 of [2] and $\overline{d}$ the equivariant bounded distance in $X$ then we define

[TABLE]

given by:

[TABLE]

Then, we extend $\overline{H}_{3}$, relative to $\partial(St(A))$, to a $G$-map $\overline{H}_{4}:X\times I\rightarrow X$. By Lemma 3.1 of [12], we eliminate the fixed points of $\overline{H}_{4}(\bullet,1)$ inside $X-St(A)\times\{1\}$. This finite set of fixed points can be removed because these fixed points lie in some non essential fixed point classes of $\overline{H}_{4}(\bullet,1)$ since $h_{2}|_{X-Int(A)}$ is fixed point free. Thus, the resulting $G$-map is a $G$-homotopy $H:X\times I\rightarrow X$ connecting $f$ to a $G$-map $h$ such that ${\rm Fix}(h)=\Phi$.

3 Proof of Theorem 1.6 - An equivariant analogue of a theorem of Brown-Soderlund

Throughout this last section, $G$ will denote a finite group. Given a $G$-fiber-preserving map $f:X\rightarrow X$ of the total space $X$ of a $G$-fiber bundle $\mathfrak{F}=(X,p,B,Y)$, it is known that the fixed point set of $f$ is related with the fixed point set of the induced map $\overline{f}:B\rightarrow B$. However, there are equivariant homotopies that are not fiber-preserving as in the example below:

Example 3.1**.**

Let $G=\mathbb{Z}_{2}$ and $X=\mathbb{S}^{2}\times\mathbb{S}^{1}$ and the action is given by $\xi((a,b,c),\cos x+i\sin x)\mapsto((a,b,c),\cos x-i\sin x)$. The $G$-map $f$, defined on $X$ by setting $f((a,b,c),\cos x+i\sin x)=((-a,-b,-c),\cos x+i\sin x)$, is the start of the following equivariant homotopy:

[TABLE]

Then, $A=\{(a,b,0)\in\mathbb{S}^{2}\}\times\{-i,i\}=\mathbb{S}^{1}\times\{i,-i\}$ is the fixed point set of $h\in[f]_{G}$ where $h=H(\bullet,1)$. Let $p=\pi_{1}:\mathbb{S}^{2}\times\mathbb{S}^{1}\rightarrow\mathbb{S}^{2}$ be the projection, then $(\mathbb{S}^{2}\times\mathbb{S}^{1},\pi_{1},\mathbb{S}^{2})$ is a $\mathbb{Z}_{2}$-fiber bundle, $f$ is a fiber-preserving map and the induced map $\overline{f}=a:\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$ is the antipodal map. However, $p((x,y,z),1)=(x,y,z)=p((x,y,z),i)$ and $p\circ h((x,y,z),1)=(-x,-y,-z)$ is different from $p\circ h((x,y,z),i)=(x,y,-z)$. So, $h$ is not a fiber preserving map and $H$ is not a fiber-preserving homotopy. In fact, $A$ cannot be realized as the fixed point set of any map equivariantly fiberwise homotopic to $f$. To see that, we note that $X^{G}=\mathbb{S}^{2}\times\{\pm 1\}=\mathbb{S}^{2}_{1}\sqcup\mathbb{S}^{2}_{-1}$, where $(w,\pm 1)\in\mathbb{S}^{2}_{\pm 1}$, consists of two disjoint $2$-spheres $\mathbb{S}^{2}$. If $F_{t}$ is a $\mathbb{Z}_{2}$ fiber-preserving homotopy such that $F_{0}=f$ and ${\rm Fix}(F_{1})=A$, then $F^{G}_{t}$ is a homotopy on $X^{G}$. Now, $f^{G}=F^{G}_{0}$ maps $\mathbb{S}^{2}_{1}$ to $\mathbb{S}^{2}_{1}$ and $\mathbb{S}^{2}_{-1}$ to $\mathbb{S}^{2}_{-1}$. On the other hand, $F_{1}$ is fiber-preserving and $A$ is the fixed point of $F_{1}$, it follows that the induced map $\overline{F_{1}}$ fixes the circle $\{(a,b,0)\in\mathbb{S}^{2}\}$ pointwise. This implies that $F_{1}$ maps the (non-fixed) point $((a,b,0),1)$ to the point $((a,b,0),-1)$ so that $F_{1}$ maps the equator of $\mathbb{S}^{2}_{1}$ to that of $\mathbb{S}^{2}_{-1}$, and vice versa. Thus $F^{G}_{1}$ maps $X^{G}$ to itself by interchangeing the two disjoint spheres $\mathbb{S}^{2}_{\pm 1}$. The images of $X^{G}$ under $F^{G}_{0}$ and $F^{G}_{1}$ contradict the continuity of $F^{G}_{t}$. Hence such an equivariant fiber-preserving homotopy $F_{t}$ cannot exist.

The example above indicates the importance of modifying the conditions $(C_{G}1)$ and $(C_{G}2)$ and replacing them by $(C_{G}1)_{\mathfrak{F}}$ and $(C_{G}2)_{\mathfrak{F}}$ for the fiber-preserving map setting.

Lemma 3.2**.**

Let $f:X\rightarrow X$ be a $G$-fiber preserving map in the total space of the $G$-fiber bundle $\mathfrak{F}=(X,p,B,Y)$, where $X$, $B$ and $Y$ are $G$-spaces $ANR$. Suppose that there is a $G$-fiber preserving homotopy connecting a $G$-fiber preserving map $h:X\rightarrow X$ to $f$ such that ${\rm Fix}(h)=A$ for a nonempty and closed $G$-subset $A$ of $X$. Then the conditions $(C_{G}1)_{\mathfrak{F}}$ and $(C_{G}2)_{\mathfrak{F}}$ given in Theorem 1.6 hold for $f$ and $A$.

The proof of Lemma 3.2 follows the steps of Lemma 2.3. Since $\mathfrak{F}=(X,p,B,Y)$ is a $G$-fiber bundle where $X$, $B$ and $Y$ are compact smooth $G$-manifolds, we observe that $(X,p,B)$ is a $G$-fibration and there is a $G$-lift map $\Lambda:\Omega_{p}\rightarrow E^{I}$ such that $\Lambda(e,\alpha)(0)=e$, $p\circ\Lambda(e,\alpha)(t)=\alpha(t)$ and $\Lambda(e,p(e))(t)=e$, for all $t\in I$, where $E^{I}=\{\alpha:I\rightarrow E;\ \alpha\ is\ a\ path\}$ and $\Omega_{p}=\{(e,\alpha)\in X\times B^{I};\ p(e)=\alpha(0)\}$.

Remark 3.3**.**

We should point out that Lemma 3.2 holds for any compact Lie group $G$ if we modify condition $(C_{G}2)_{\mathfrak{F}}$ by only considering those $(K)$’s with $|WK|<\infty$.

The next proposition is an equivariant analogue of Theorem 2.1 of [1].

Proposition 3.4**.**

Let $\overline{H}:(X\times\{0\})\cup(A\times I)\rightarrow E$ be a $G$-map in the $G$-fibration $\mathfrak{F}=(E,p,B)$, where $E$ is a $G$-ANR, $A$ is a closed $G$-subset of $X$, $(X,A)$ is a $G$-metric pair and $p\circ\overline{H}(x,0)=p\circ{\overline{H}}(x,t)$ for all $(x,t)\in A\times I$. Then $\overline{H}$ can be extended to a $G$-homotopy $H:X\times I\rightarrow E$ such that $p\circ H(x,0)=p\circ H(x,t)$ for all $(x,t)\in X\times I$.

Proof of Proposition 3.4: Let $H^{\prime}:X\times I\rightarrow E$ a $G$-extension of $\overline{H}$. Then $H^{\prime}$ is given by:

[TABLE]

Then define $H(x,t)=\Lambda(H^{\prime}(x,t),p(H^{\prime}(x,\bullet))_{t})(1)$, where $p(H^{\prime}(x,\bullet))_{t}(s)=p(H^{\prime}(x,(1-s)t))$ and $\Lambda$ is a $G$-lift map.

$\Box$

Lemma 3.5**.**

Let $\mathfrak{F}=(X,p,B,Y)$ be a $G$-fiber bundle where $X,B$ and $Y$ are compact and smooth $G$-manifolds, $\dim(B^{K})\geq 3$, $\dim(B^{K})-\dim(B^{K}-B_{K})\geq 2$, for all $(K)\in{\rm Iso}(B)$, $A$ be a nonempty, closed, locally contractible $G$-subset of $X$ such that $p(A)$ be a closed $G$-subset of $B$ and $p^{K}(A^{K})$ is by-passed in $B^{K}$, for all $(K)\in{\rm Iso}(B)$, and $f:X\rightarrow X$ a $G$- fiber preserving map such that conditions $(C_{G}1)_{\mathfrak{F}}$ and $(C_{G}2)_{\mathfrak{F}}$ given in Theorem 1.6 hold for $f$ and $A$.

Then there exists a $G$-fiber-preserving map $h$, $G$-fiberwise homotopic to $f$ with $A\subset{\rm Fix}(h)\subset p^{-1}(p(A))$ and ${\rm Fix}(\overline{h})\cap(B-p(A))$ is a finite set.

Proof of Lemma 3.5: $p(A)$ is a closed $G$-subset of $B$ then the $G$-fiber-preserving map $H_{A}:A\times I\rightarrow X$ given by $(C_{G}1)_{\mathfrak{F}}$ induces a $G$-map $\overline{H}_{A}:p(A)\times I\rightarrow B$ such that $\overline{H}_{A}(\bullet,0)=\overline{f}$ and $\overline{H}_{A}(\bullet,1)=i_{p(A)}:p(A)\hookrightarrow B$ the inclusion map.

Observe that we have almost the same conditions that we had in Theorem 1.5 except for $(C_{G}2)$. In this situation, suppose we have a $G$-map $\overline{H}_{i-1,A}:(p(A)\cup B_{i-1})\times I\rightarrow B$. As commented in Lemma 2.5, it is possible to extend $\overline{H}_{i-1,A}$ to a $G$-map $\overline{H}_{i,1}:(B_{i}\cup p(A))\times I\rightarrow B$ relative to $p(A)\cup B_{i-1}$.

Since $WK_{i}$ is a finite group, $B_{i}^{K_{i}}$ is a $WK_{i}$- polyhedron such that $B_{i-1}^{K_{i}}$ is a $WK_{i}$-subpolyhedron of $B_{i}^{K_{i}}$ and $St(p(A_{i}^{K_{i}}))$ is neighborhood by-passed in $B_{i}^{K_{i}}$. Let $V$ be a $G$-invariant neighborhood retract of $St(p(A_{i}))\cup B_{i-1}$. It follows from Lemma 3.1 of [12] and Lemma 2.1 that there exists of a $WK_{i}$-homotopy $\overline{H}_{i}:B_{i}^{K_{i}}\times I\rightarrow B_{i}^{K_{i}}$ from $\overline{H}_{i,1}^{K_{i}}(\bullet,1)$ to $\overline{h}=\overline{H}_{i}(\bullet,1)$ such that:

$p(A)^{K_{i}}_{i}\subset{\rm Fix}(\overline{h})$; 2. 2.

$\overline{h}$ has a finite number of fixed points in $B^{K_{i}}_{i}-V^{K_{i}}$; 3. 3.

given a $WK_{i}$-fixed point class $F$ of $\overline{h}$ such that $F\cap p(A)^{K_{i}}_{i}=\emptyset$ then $F=WK_{i}\{x\}$, where $x\in B^{K_{i}}_{i}-V^{K_{i}}$ and $F$ is an essential $WK_{i}$-fixed point class of $\overline{h}$.

Then, the $G$-map given by:

[TABLE]

extends a $WK_{i}$-homotopy to a $G$-homotopy $\overline{H}_{i}:(B_{i}\cup p(A))\times I\rightarrow B$ relative to $V$ and such that

[TABLE]

and $WK_{i}\{b_{i,l}\}$ is a essential $WK_{i}$-fixed point class of $\overline{H}_{i}^{K_{i}}(\bullet,1)$, for $1\leq l\leq m_{i}$.

Observe that if $p^{K}(F)=WK\{b_{i,l}\}$ for an essential $WK$-fixed point class $F$ of $f^{K}$ where $(K)\in{\rm Iso}(X)$, then we have a path $\overline{\alpha}$ such that:

[TABLE]

Hence, $\overline{\alpha}(1)=gb_{i,l}$, for some $g\in WK$ and $\overline{\alpha}(1)\in p^{K}(A)$. However, this cannot occur because $b_{i,l}\notin p^{K}(A)$ and $p^{K}(A)$ is $WK$-invariant. By induction we extend the $G$-map $\overline{H}_{A}:p(A)\times I\rightarrow B$ to a $G$-homotopy $\overline{H}:B\times I\rightarrow B$ with the properties above.

Note that $H^{\prime}:X\times I\rightarrow B$ defined by $H^{\prime}(x,t)=\overline{H}(p(x),t)$ is such that

[TABLE]

Therefore, the lift of $H^{\prime}$ is a fiber-preserving $G$-homotopy $H_{1}:X\times I\rightarrow X$ such that $f(x)=H_{1}(x,0)$ and $h_{1}(x)=H_{1}(x,1)$. Thus,

[TABLE]

For each $G$-orbit $G\{b_{j}\}$ take the restriction $h_{1,b_{j}}$ of $h_{1}$ for $Gp^{-1}(b_{j})=p^{-1}(G\{b_{j}\})$, so $h_{1,b_{j}}:Gp^{-1}(b_{j})\rightarrow Gp^{-1}(b_{j})$ has no essential fixed point classes. In fact, suppose that $h_{1,b_{j}}^{K}$ has an essential $WK$-fixed point class $F$. Then, given $x\in F$ we have $WK\{x\}$ lying inside an essential $WK$-fixed point class of $h_{1}^{K}$. Thus, there exists a $WK$-fixed point class $Q$ of $h_{1}^{K}$ which contains $WK\{x\}$. But, $h_{1}^{K}$ is fiber-preserving $WK$-homotopic to $f^{K}$, so, there exists an essential $WK$-fixed point class $D$ of $f^{K}$ $H_{1}^{K}$-related to $Q$. Note that $p^{K}(D)$ cannot be $\overline{H}^{K}$-related to $WK\{b_{j}\}$. Consequently, $h_{1,b_{j}}$ is fiber-preserving $G$-homotopic to $h_{2,b_{j}}:Gp^{-1}(b_{j})\rightarrow Gp^{-1}(b_{j})$ fixed point free.

Consider the $G$-map

[TABLE]

defined by:

[TABLE]

With Proposition 3.4 we extend $\widetilde{H}_{2}$ to a fiber-preserving $G$-homotopy $H_{2}:X\times I\rightarrow X$ and $h_{2}=H_{2}(\bullet,1)$ is such that $\overline{H}_{2}(\bullet,1)=\overline{h}$. By $(C_{G}1)_{\mathfrak{F}}$, $h|_{2}$ is fiber-preserving $G$-homotopic to $i_{A}$. Let $\widetilde{H}_{A}$ such that $h_{2}|_{A}=\widetilde{H}_{A}(\bullet,0)$ and $i_{A}=\widetilde{H}_{A}(\bullet,1)$. Define $\widetilde{H}:(X\times\{0\})\cup((A\cup p^{-1}(G\{b_{1},\dots,b_{r}\}))\times I)\rightarrow X$ given by:

[TABLE]

Applying Proposition 3.4 again we extend $\widetilde{H}$ to a fiber-preserving $G$- homotopy $H:X\times I\rightarrow X$ such that $A\subset{\rm Fix}(h)\subset p^{-1}(p(A))$ and ${\rm Fix}(\overline{h})\cap(B-p(A))$ is a finite set.

$\Box$

Lemma 3.6**.**

Let $(\mathfrak{F},\mathfrak{F}_{0})=((X,A),p,B,(Y,Y_{0}))$ be a $G$-fiber bundle pair, where $X$, $B$ and $Y$ are compact and smooth $G$-manifolds, $B$ retracts equivariantly to a point $b_{0}\in B$ and $\dim(Y^{K})\geq 3$ and $\dim(Y^{K})-\dim(Y^{K}-Y_{K})\geq 2$, for all $(K)\in{\rm Iso}(Y)$. Let $Y_{0}$ be a closed and locally contractible $G$-subset of $Y$ such that $Y^{K}_{0}$ is by-passed in $Y^{K}$, for all $(K)\in{\rm Iso}(Y)$, $A$ be a nonempty, closed, locally contractible $G$-subset of $X$ and $f:X\rightarrow X$ be a $G$-map such that $p\circ f=p$, $A\subset{\rm Fix}(f)$, $A^{K}$ intersects every essential $WK$-fixed point class of $f_{b_{0}}^{K}:WK(p^{K})^{-1}(\{b_{0}\})\rightarrow WK(p^{K})^{-1}(\{b_{0}\})$, for all $(K)\in{\rm Iso}(X)$.

Then for every closed $G$-invariant subset $Z$ of $A$ that intersects every component of $A$ and $(A,Z)$ is $G$-fiber bundle pair of $\mathfrak{F}_{0}$ there exists a fiber-preserving $G$-map $h$, $G$-fiberwise homotopic to $f$ with ${\rm Fix}(h)=Z$.

Proof of Lemma 3.6: $(X,p,B)$ is $G$-equivalent to a trivial $G$-fibration $(B\times Y,\pi,B)$, where $\pi$ is a projection in $B$. So, there exists a $G$- homeomorphism $\Phi:B\times Y\rightarrow X$ such that $\Phi(B\times Y_{0})=A$ and $p\circ\Phi=\pi$. Define $f^{*}=\Phi^{-1}\circ f\circ\Phi:B\times Y\rightarrow B\times Y$ and note that:

[TABLE]

Therefore, $f^{*}(b,y)=(b,f^{*}_{b}(y))$ and $gf^{*}(b,y)=(gb,f^{*}_{gb}(gy))$, for all $g\in G$.

$B$ retracts equivariantly to $b_{0}$, so, there exists a $G$-homotopy $D:B\times I\rightarrow B$ such that for each $b\in B$ we have $D(b,0)=b$ and $D(b,1)=b_{0}$. Then, define $U^{*}:B\times Y\times I\rightarrow B\times Y$ given by:

[TABLE]

Note that $U^{*}(b,y,0)=(b,f^{*}_{b}(y))=f^{*}(b,y)$ and $U^{*}(b,y,1)=(b,f^{*}_{b_{0}}(y))$. Then, $f^{*}$ is $G$-homotopic to $id\times f^{*}_{b_{0}}$. Since $\Phi(B\times Y_{0})=A\subset{\rm Fix}(f)$ we have, for each $(b,y)\in B\times Y_{0}$:

[TABLE]

Then, $B\times Y_{0}\subset{\rm Fix}(f^{*})$ and $Y_{0}\subset{\rm Fix}(f^{*}_{b})$ because $(b,f^{*}_{b}(y))=f^{*}(b,y)=(b,y)$.

By hypothesis, $A$ intersects each essential $WK$-fixed point class of $f^{K}_{b_{0}}:WK(p^{K})^{-1}(b_{0})\rightarrow WK(p^{K})^{-1}(b_{0})$. So, $A\cap WK(p^{K})^{-1}(b_{0})$ intersects each essential $WK$-fixed point class of $f_{b_{0}}^{K}$. So, $Y_{0}$ intersects each essential $WK$-fixed point class of $(f^{*}_{b_{0}})^{K}$ because:

[TABLE]

The $G$-fiber bundle pair \Big{(}(A,Z),p,B,(Y_{0},\Omega)\Big{)} is such that $\Omega$ intersects every component of $Y_{0}$ because $Z$ intersects every component of $A$. Therefore, $(C_{G}1)_{\mathfrak{F}}$ and $(C_{G}2)_{\mathfrak{F}}$ hold for $Y_{0}$ and $f^{*}_{b_{0}}$. By Theorem 2.2 there exists a homotopy $V^{*}:Y\times I\rightarrow Y$ such that $f^{*}_{b_{0}}=V^{*}(\bullet,0)$, $V^{*}(\bullet,1)=g^{*}_{b_{0}}:Y\rightarrow Y$ and ${\rm Fix}(g^{*}_{b_{0}})=\Omega$. Define a fiber-preserving $G$-homotopy $H^{*}:B\times Y\times I\rightarrow B\times Y$ given by:

[TABLE]

Therefore, $f^{*}$ is $G$-homotopic to $id\times g^{*}_{b_{0}}$ and ${\rm Fix}(id\times g^{*}_{b_{0}})=B\times\Omega$. Then, $H:X\times I\rightarrow X$ given by $H(e,t)=\Phi\circ H^{*}(\Phi^{-1}(e),t)$ is a $G$-homotopy such that $H(x,0)=f(x)$ and ${\rm Fix}(H(\bullet,1))=\Phi(B\times\Omega)=Z$.

$\Box$

Proof of Theorem 1.6: With Lemma 3.5 we assume that:

$A\subset{\rm Fix}(f)\subset p^{-1}(p(A))$; 2. 2.

$F={\rm Fix}(\overline{f})\cap(B-p(A))$ is a finite set.

Let $f_{j}=f|_{p^{-1}(p(A)_{j})}:p^{-1}(p(A)_{j})\rightarrow p^{-1}(p(A)_{j})$ a restriction of $f$, so $p\circ f_{j}=p$. Using $X=p^{-1}(p(A)_{j}),\ A=A\cap p^{-1}(p(A)_{j}),\ B=p(A)_{j},\ b_{0}=b_{j},\ Y_{0}=Y_{j}$ and $f=f_{j}$ the hypotheses of Lemma 3.6 are satisfied and there exists a fiber-preserving $G$-homotopy $H_{j}:p^{-1}(p(A)_{j})\times I\rightarrow p^{-1}(p(A)_{j})$ from $f_{j}$ to $h_{j}=H_{j}(\bullet,1)$ such that ${\rm Fix}(h_{j})=Z_{j}$.

Define $\widetilde{H}_{2}:(X\times\{0\})\cup(p^{-1}(F\cup p(A))\times I)\rightarrow X$ by:

[TABLE]

With Proposition 3.4 there is a fiber-preserving $G$-homotopy $H:X\times I\rightarrow X$ such that $p(H(x,t))=\overline{f}\circ p(x)$. Therefore, $h=H(\bullet,1):X\rightarrow X$ is such that ${\rm Fix}(h)=Z$ and $h$ is fiber-preserving $G$-homotopic to $f$.

$\Box$

Corollary 3.7**.**

Let $\mathfrak{F}=(X,p,B,Y)$ be a $G$-fiber bundle where $X$, $B$ and $Y$ are compact and smooth $G$-manifolds, $\dim(B^{K})\geq 3$, $\dim(B^{K})-\dim(B^{K}-B_{K})\geq 2$, for all $(K)\in{\rm Iso}(B)$, $\dim(Y^{K})\geq 3$, $\dim(Y^{K})-\dim(Y^{K}-Y_{K})\geq 2$, for all $(K)\in{\rm Iso}(Y)$.

Let $A$ be a nonempty, closed, locally contractible $G$-subset of $X$ such that $(X,A)$ is $G$-fiber bundle pair with respect to the fiber bundle $\mathfrak{F}$, $p(A)$ be a closed $G$-subset of $B$ such that each component $p(A)_{j}$ of $p(A)$ is equivariantly contractible and $p^{K}(A^{K})$ is by-passed in $B^{K}$, for all $(K)\in{\rm Iso}(B)$. Let $Y_{j}$ be a subbundle fiber of $A$ such that $Y_{j}$ is a closed and locally contractible $G$-subset of $Y$ and $Y^{K}_{j}$ is by-passed in $Y^{K}$, for all $(K)\in{\rm Iso}(Y)$, and $f:X\rightarrow X$ be a $G$-fiber-preserving map such that $A^{K}$ intersects every essential $WK$-fixed point class of $f^{K}_{b_{j}}:WK(p^{K})^{-1}(\{b_{j}\})\rightarrow WK(p^{K})^{-1}(\{b_{j}\})$ for at least one $b_{j}$ in each component $p^{K}(A^{K})_{j}$, for all $(K)\in{\rm Iso}(X)$.

Then there exists a $G$-fiber-preserving map $h$, $G$-fiberwise homotopic to $f$ with ${\rm Fix}(h)=A$ if, and only if, the following conditions holds for $f$ and $A$:

• $(C_{G}1)_{\mathfrak{F}}$

there exists a $G$-fiber-homotopy $H_{A}:A\times I\rightarrow X$ from $f|_{A}$ to the inclusion $i:A\hookrightarrow X$;

• $(C_{G}2)_{\mathfrak{F}}$

for every $WK$-essential fixed point class $F$ of $f^{K}:X^{K}\rightarrow X^{K}$ there exists a path $\alpha:I\rightarrow X^{K}$ with $\alpha(0)\in F$, $\alpha(1)\in A^{K}$, and $\{\alpha(t)\}\sim\{f^{K}\circ\alpha(t)\}\ast\{H_{A}^{K}(\alpha(1),t)\}.$

Proof of Corollary 3.7: If the conditions hold then we apply Theorem 1.6 for $Z=A$. If there exists $h$ then by Lemma 3.2 $(C_{G}1)_{\mathfrak{F}}$ and $(C_{G}2)_{\mathfrak{F}}$ hold.

$\Box$