Equivariant formality and the cohomology of subgroups of right-angled Coxeter groups

TL;DR

This paper models the classifying spaces of certain subgroups of right-angled Coxeter groups using homotopy orbit spaces, linking their cohomology to Borel equivariant cohomology and providing a graph-theoretic criterion for cohomology freeness.

Contribution

It generalizes models for classifying spaces of right-angled Coxeter groups to coabelian subgroups and characterizes equivariant formality via a simple graph criterion.

Findings

Models for classifying spaces as homotopy orbit spaces of real moment-angle complexes.

Cohomology identified with Borel equivariant cohomology of elementary abelian 2-group actions.

Graph-theoretic criterion for cohomology to be free over the quotient's cohomology.

Abstract

We construct models for the classifying spaces of coabelian subgroups of right-angled Coxeter groups as homotopy orbit spaces of real moment-angle complexes, generalizing well-known models for the classifying space of a right-angled Coxeter group and its commutator subgroup. This identifies the cohomology of these groups with the Borel equivariant cohomology of elementary abelian $2$-group actions on cubical subcomplexes of a cube $[-1,1]^m$. We then characterize equivariant formality for these actions, leading to a simple graph-theoretic criterion for when the cohomology of a coabelian subgroup is free as a module over the cohomology of the quotient by the commutator subgroup of the right-angled Coxeter group.