Representation-graded Bredon homology of elementary abelian 2-groups

TL;DR

This paper computes the representation-graded Bredon homology rings for all elementary abelian 2-groups with mod-2 coefficients, providing explicit presentations and applications to fixed point and twisted cohomology rings.

Contribution

It offers the first explicit minimal presentations of these homology rings as quotients of polynomial algebras with relations, extending previous calculations.

Findings

Explicit minimal presentations of the rings as polynomial quotients

Calculation of geometric fixed point rings

Strengthening of localized twisted cohomology ring calculations

Abstract

We calculate the representation-graded Bredon homology rings of all elementary abelian 2-groups with coefficients in the constant mod-2 Mackey functor. We exhibit minimal presentations for these rings as quotients of the polynomial algebra on the pre-Euler and inverse Thom classes of all nontrivial characters, subject to an explicit finite list of relations arising from orientability properties. Two corollaries of our presentation are the calculation, originally due to Holler and Kriz, of the geometric fixed point rings, and a strengthening of a calculation of Balmer and Gallauer of the localized twisted cohomology ring.