Torus actions whose equivariant cohomology is Cohen-Macaulay

Details Torus actions whose equivariant cohomology is Cohen-Macaulay Torus actions whose equivariant cohomology is Cohen-Macaulay

2009-12-03

arxiv.org/abs/0912.0637v2

J. Topology (2010) 3(4) 819-846

Mathematics

Algebraic Topology

28 pages. V2 contains changes according to the referee's suggestions

Scientific paper

10.1112/jtopol/jtq025

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.

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