Representation-graded Bredon homology of elementary abelian 2-groups
Markus Hausmann, Stefan Schwede
TL;DR
This paper determines the representation-graded Bredon homology rings $H(A,\\star)$ for all elementary abelian $2$-groups $A$ with constant mod-2 coefficients, giving a minimal presentation as a quotient of the polynomial algebra on the pre-Euler classes $a_\\lambda$ and inverse Thom classes $t_\\lambda$ for all nontrivial $A$-characters $\\lambda$, subject to relations $r(T)=\\sum_{\\lambda\\in T} a_\\lambda\\cdot\\prod_{\\mu\\in T\\setminus\\{\\lambda\\}} t_\\mu$ for minimally dependent sets $T$ of characters. It is shown that $H(A,\\star)$ is a domain, and two corollaries are derived: (i) the geometric fixed point ring $\\Phi^A_*(H\\underline{\\mathbb F}_2)$ obtained by inverting all $a_\\lambda$, reproducing Holler and Kriz's calculation, and (ii) a strengthened localization $H(A|B)$ generalizing Balmer and Gallauer's twisted cohomology, with explicit generators and relations. The results yield a global universal property: the system of representation-graded Bredon homology rings forms an initial additively oriented $el^{RO}_2$-algebra in the mod-2 setting. The work connects to classical computations and clarifies how localization simplifies the relation algebra, including rank-3 examples and a combinatorial count of minimally dependent sets.
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.