Quantifying Quillen’s uniform $\mathcal{F}_p$-isomorphism theorem

Let $G$ be a finite group with $2$-Sylow subgroup of order less than or equal to $16$. For such a $G$, we prove a quantified version of Quillen’s uniform $\mathcal{F}_p$-isomorphism theorem, which holds uniformly for all $G$-spaces.

We do this by bounding from above the exponent of Borel equivariant $\mathbf{F}_2$-cohomology, as introduced by Mathew–Naumann–Noel, with respect to the family of elementary abelian $2$-subgroups.

Keywords

group cohomology, Quillen’s F-isomorphism theorem, equivariant homotopy theory, spectral sequence

2010 Mathematics Subject Classification

18G40, 20J06, 55N91, 55P42, 55P91

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The author was partly supported by the SFB 1085 – Higher Invariants, Regensburg.

This article was revised on June 29, 2022 to correct the names used for internal cross-references.

Received 6 January 2018

Received revised 3 October 2019

Published 25 March 2020