Obstructions for constructing equivariant fibrations

Abstract

Let $G$ be a finite group and $\mathcal{\mathscr{H}}$ 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 $\mathcal{\mathscr{H}}$ and let $\mathcal{ℱ}={\left\{F_H\right\}}_{H\in \mathcal{\mathscr{H}}}$ be a compatible family of $H$–spaces. A $G$–fibration over $B$ with the fiber type $\mathcal{ℱ}={\left\{F_H\right\}}_{H\in \mathcal{\mathscr{H}}}$ is a $G$–equivariant fibration $p:E\to B$ where $p^{-1}\left(b\right)$ 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 the fiber type $\mathcal{ℱ}$ over a given $G$–CW–complex $B$. Constructing $G$–fibrations with a prescribed fiber type $\mathcal{ℱ}$ is an important step in the construction of free $G$–actions on finite CW–complexes which are homotopy equivalent to a product of spheres.