Dimension functions for spherical fibrations

Okay, Cihan; Yalçin, Ergün

doi:10.2140/agt.2018.18.3907

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Dimension functions for spherical fibrations

Cihan Okay and Ergün Yalçin

Algebraic & Geometric Topology 18 (2018) 3907–3941

DOI: 10.2140/agt.2018.18.3907

Abstract

Given a spherical fibration $ξ$ over the classifying space $B G$ of a finite group $G$ we define a dimension function for the $m$ –fold fiber join of $ξ$ , 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 $Qd (p) = {(ℤ ∕ p)}^{2} ⋊ {SL}_{2} (ℤ ∕ p)$ 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

References

Publication

Received: 30 October 2017

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

Volume 18, issue 7 (2018)