Torus actions whose equivariant cohomology is Cohen-Macaulay

TL;DR

This paper investigates Cohen-Macaulay torus actions on manifolds, providing conditions for their equivariant cohomology to be computable and exploring their properties in relation to fixed points and orbit structures.

Contribution

It introduces sufficient conditions for Cohen-Macaulay properties in torus actions, extending the understanding of equivariant cohomology beyond fixed point sets.

Findings

Cohen-Macaulay actions generalize equivariantly formal actions.

Chang-Skjelbred Lemma and Atiyah-Bredon sequence are applicable.

Conditions like invariant Morse-Bott functions ensure Cohen-Macaulay property.

Abstract

We study Cohen-Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang-Skjelbred Lemma, and its stronger version, the exactness of an Atiyah-Bredon sequence, hold. The main difference is that the fixed point set is replaced by the union of lowest dimensional orbits. We find sufficient conditions for the Cohen-Macaulay property such as the existence of an invariant Morse-Bott function whose critical set is the union of lowest dimensional orbits, or open-face-acyclicity of the orbit space. Specializing to the case of torus manifolds, i.e., 2r-dimensional orientable compact manifolds acted on by r-dimensional tori, the latter is similar to a result of Masuda and Panov, and the converse of the result of Bredon that equivariantly formal torus manifolds are open-face-acyclic.