# 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.

## Key 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.