Torsion in equivariant cohomology and Cohen-Macaulay G-actions

TL;DR

This paper generalizes the relationship between torsion-free equivariant cohomology and injective fixed point maps from torus actions to actions of arbitrary compact connected Lie groups, introducing new conditions involving isotropy rank and Cohen-Macaulay properties.

Contribution

It extends classical results on equivariant cohomology to more general Lie group actions using isotropy rank and Cohen-Macaulay conditions, and provides a topological criterion for equivariant injectivity.

Findings

Equivariant cohomology is torsion-free iff the induced map from fixed points is injective for torus actions.

For arbitrary compact Lie groups, the fixed point set condition is replaced by points with maximal isotropy rank.

The action on the set of points with highest isotropy rank is Cohen-Macaulay.

Abstract

We show that the well-known fact that the equivariant cohomology of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with maximal isotropy rank. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.