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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math>$ over the classifying space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mi>G</mi></math>$ of a finite group $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math>$ we define a dimension function for the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math>$ –fold fiber join of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math>$ , where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math>$ is some large positive integer. We show that the dimension functions satisfy the Borel–Smith conditions when $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math>$ is large enough. As an application we prove that there exists no spherical fibration over the classifying space of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Qd</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>p</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">=</mo> <msup><mrow><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℤ</mi><mo class="MathClass-bin">∕</mo><mi>p</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup> <mo class="MathClass-bin">⋊</mo><msub><mrow><mo class="qopname"> SL</mo></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℤ</mi><mo class="MathClass-bin">∕</mo><mi>p</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi></math>$ –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)