Dimension functions for spherical fibrations

Abstract

Given a spherical fibration $\hat{I}¾$ over the classifying space $BG$ of a finite group we define a dimension function for the $m-$fold fiber join of $\hat{I}¾$ 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 $\text{Qd}(p)=(\mathbb{Z}/p{)}^2⋊{\text{SL}}_2(\mathbb{Z}/p)$ with $p-$effective Euler class, generalizing the result of Özgün Ünlü about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in a future paper as a corollary of a previously announced program on homotopy group actions due to Jesper Grodal.