#### Volume 18, issue 7 (2018)

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Dimension functions for spherical fibrations

### Cihan Okay and Ergün Yalçin

Algebraic & Geometric Topology 18 (2018) 3907–3941
##### Abstract

Given a spherical fibration $\xi$ over the classifying space $BG$ of a finite group $G$ we define a dimension function for the $m$–fold fiber join of $\xi$, where $m$ is some large positive integer. We show that the dimension functions satisfy the Borel–Smith conditions when $m$ is large enough. As an application we prove that there exists no spherical fibration over the classifying space of $Qd\left(p\right)={\left(ℤ∕p\right)}^{2}⋊{SL}_{2}\left(ℤ∕p\right)$ with $p$–effective Euler class, generalizing a result of Ünlü (2004) about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in upcoming work of Alejandro Adem and Jesper Grodal as a corollary of a previously announced program on homotopy group actions due to Grodal.

##### Keywords
group actions, Smith theory, spherical fibrations, Lannes' $T$–functor
##### Mathematical Subject Classification 2010
Primary: 55M35
Secondary: 55S10, 55S37
##### Publication
Revised: 1 June 2018
Accepted: 11 June 2018
Published: 11 December 2018
##### Authors
 Cihan Okay Department of Mathematics Middlesex College Western University London, ON Canada Ergün Yalçin Department of Mathematics Bilkent University Ankara Turkey