Obstructions for constructing equivariant fibrations

Details Obstructions for constructing equivariant fibrations Obstructions for constructing equivariant fibrations

2011-10-18

arxiv.org/abs/1110.3880v1

Mathematics

Algebraic Topology

Scientific paper

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.

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