Representation-graded Bredon homology of elementary abelian 2-groups

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.