Syzygies in equivariant cohomology for non-abelian Lie groups

TL;DR

This paper generalizes the theory of syzygies in equivariant cohomology from tori to all compact connected Lie groups, establishing new criteria for exact sequences and Cohen-Macaulay modules.

Contribution

It extends syzygy theory to non-abelian Lie groups, introduces a Cartan model for singular spaces, and provides criteria for orbit type finiteness.

Findings

Exactness of the Chang-Skjelbred sequence characterizes reflexive equivariant cohomology.

Equivariant cohomology modules from orbit filtrations are Cohen-Macaulay.

Develops a criterion for finiteness of infinitesimal orbit types.

Abstract

We extend the work of Allday-Franz-Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincare pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen-Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We also develop a criterion for the finiteness of the number of infinitesimal orbit types of a G-manifold.