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Dimension functions for spherical fibrations

Cihan Okay and Ergün Yalçin

Algebraic & Geometric Topology 18 (2018) 3907–3941

Given a spherical fibration ξ over the classifying space BG 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) = (p)2 SL2(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.

group actions, Smith theory, spherical fibrations, Lannes' $T$–functor
Mathematical Subject Classification 2010
Primary: 55M35
Secondary: 55S10, 55S37
Received: 30 October 2017
Revised: 1 June 2018
Accepted: 11 June 2018
Published: 11 December 2018
Cihan Okay
Department of Mathematics
Middlesex College
Western University
London, ON
Ergün Yalçin
Department of Mathematics
Bilkent University