#### Volume 12, issue 3 (2012)

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Obstructions for constructing equivariant fibrations

### Aslı Güçlükan İlhan

Algebraic & Geometric Topology 12 (2012) 1313–1330
##### Abstract

Let $G$ be a finite group and $\mathsc{ℋ}$ 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 $\mathsc{ℋ}$ and let $\mathsc{ℱ}={\left\{{F}_{H}\right\}}_{H\in \mathsc{ℋ}}$ be a compatible family of $H$–spaces. A $G$–fibration over $B$ with the fiber type $\mathsc{ℱ}={\left\{{F}_{H}\right\}}_{H\in \mathsc{ℋ}}$ 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 $\mathsc{ℱ}$ over a given $G$–CW–complex $B$. Constructing $G$–fibrations with a prescribed fiber type $\mathsc{ℱ}$ is an important step in the construction of free $G$–actions on finite CW–complexes which are homotopy equivalent to a product of spheres.

##### Keywords
equivariant fibration, Bredon cohomology, obstruction theory, group action
Primary: 57S25
Secondary: 55R91