Obstructions for constructing equivariant fibrations

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 $p:E\to B$ 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.

References in corpus (1)

• Constructing free actions of p-groups on products of spheres

Cited by in corpus (1)

• Equivariant parametrized topological complexity