Some computations with the $\mathscr{F}$-homotopy limit spectral sequence

TL;DR

This paper provides explicit bounds on the nilpotence degree and power needed for elements in the Borel equivariant cohomology to be in the image or kernel of the comparison map, focusing on groups of order ≤ 16.

Contribution

It offers explicit upper and lower bounds for the $ ext{E}_2$-exponent of the spectral sequence for small 2-groups, extending the uniform $ ext{F}_p$-isomorphism theorem to these cases.

Findings

Kernel elements have nilpotence degree ≤ 4 for groups of order ≤ 16.

Elements in the codomain raised to the 8th power are in the image for these groups.

Bounds are established by analyzing the homotopy limit spectral sequence.

Abstract

The uniform $\mathcal{F}_p$-isomorphism theorem of Quillen gives a comparison map between the Borel equivariant $\mathbf{Z}/p$-cohomology of a space and a limit involving only the Borel equivariant cohomology groups of the same space with the action restricted to the elementary abelian $p$-subgroups. The theorem states that the elements in the kernel are nilpotent, and that every element in the codomain has a power that is in the image. By work of Mathew-Naumann-Noel, for a fixed group and an arbitrary space, there is a uniform bound on the nilpotence degree of the elements in the kernel, and on the power necessary to raise an element in the codomain by to get an element in the image. Using their methods, we give for $p=2$ explicit upper bounds for finite groups with 2-Sylow of order less than or equal to 16. In particular, we show that the elements in the kernel have in that case nilpotence degree less than or equal to 4, and every element in the codomain raised to the power 8 is in the image. We do this by bounding from above the $\mathscr{E}_{(2)}$-exponent, as defined by Mathew-Naumann-Noel, of Borel equivariant singular $\mathbf{Z}/2$-cohomology for the 2-groups of order less than or equal to 16. We also bound the exponent from below by evaluating a homotopy limit spectral sequence for all these groups.