Dimension functions for spherical fibrations

TL;DR

This paper introduces dimension functions for spherical fibrations over classifying spaces of finite groups, demonstrating they satisfy Borel-Smith conditions and applying this to prove nonexistence results for certain fibrations related to group actions.

Contribution

It defines new dimension functions for spherical fibrations and proves their properties, leading to a generalization of nonexistence results for spherical fibrations with specific Euler classes.

Findings

Dimension functions satisfy Borel-Smith conditions for large m

Nonexistence of certain spherical fibrations over classifying spaces of Qd(p)

Extension of previous results on group actions on spheres

Abstract

Given a spherical fibration $\xi$ over the classifying space $BG$ of a finite group 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 $\text{Qd}(p)= (\mathbb{Z}/p)^2\rtimes\text{SL}_2(\mathbb{Z}/p)$ with $p-$effective Euler class, generalizing the result of \"Ozg\"un \"Unl\"u 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.