Obstructions for constructing equivariant fibrations

TL;DR

This paper develops an obstruction theory to determine when it is possible to construct equivariant fibrations with prescribed fibers over a $G$-CW complex, aiding in the construction of free group actions on complexes homotopy equivalent to products of spheres.

Contribution

It introduces an obstruction-theoretic framework for constructing $G$-fibrations with specific fibers over $G$-CW complexes, advancing the understanding of equivariant fiber bundle construction.

Findings

Provides criteria for the existence of $G$-fibrations with given fibers.

Develops a systematic obstruction theory for equivariant fibrations.

Facilitates the construction of free $G$-actions on complexes homotopy equivalent to products of spheres.

Abstract

Let $G$ be a finite group and $\mathcal{H}$ be a family of subgroups of $G$ which is closed under conjugation and taking subgroups. Let $B$ be a $G$-$CW$-complex whose isotropy subgroups are in $\mathcal{H}$ and let $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ be a compatible family of $H$-spaces. A $G$-fibration over $B$ with fiber $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ is a $G$-equivariant fibration $p:E \rightarrow B$ where $p^{-1}(b)$ is $G_b$-homotopy equivalent to $F_{G_b}$ for each $b \in B$. In this paper, we develop an obstruction theory for constructing $G$-fibrations with fiber $\mathcal{F} $ over a given $G$-$CW$-complex $B$. Constructing $G$-fibrations with a prescribed fiber $\mathcal{F}$ is an important step in the construction of free $G$-actions on finite $CW$-complexes which are homotopy equivalent to a product of spheres.