Dimension functions for spherical fibrations

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 $Qd\left(p\right)={\left(\mathbb{Z}∕p\right)}^2⋊{SL}_2\left(\mathbb{Z}∕p\right)$ 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.