Quantifying Quillen's Uniform $\mathcal{F}_p$-isomorphism Theorem

TL;DR

This paper provides a quantified version of Quillen's uniform $_p$-isomorphism theorem for finite groups with small 2-Sylow subgroups, by bounding the exponent of Borel equivariant cohomology uniformly across all such groups.

Contribution

It introduces a quantitative bound on the exponent of Borel equivariant $_2$-cohomology for finite groups with small 2-Sylow subgroups, extending Quillen's theorem.

Findings

Established a uniform bound for the exponent of Borel equivariant cohomology.

Proved the quantified version for groups with 2-Sylow subgroup order ≤ 16.

Extended the applicability of Quillen's theorem to a broader class of groups.

Abstract

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.