Torus actions whose equivariant cohomology is Cohen-Macaulay

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, that is, 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.

Equivariant cohomology of cohomogeneity one actions

“…The relevance of this notion for the theory of equivariant cohomology was for the first time noticed by Bredon in [9]. Other references are [16], [18], and [19].…”

“…To find its dimension, we consider the identity component K 0 of K and note that also H * (BK 0 ) is a Cohen-Macaulay module over H * (BG); by Lemma 2.5, the dimension of H * (BK 0 ) over H * (BG) is equal to the (Krull) dimension of the (polynomial) ring H * (BK 0 ), which is rank K 0 . Let us now observe that H * (BK) = H * (BK 0 ) K/K0 is a nonzero H * (BG)-submodule of the Cohen-Macaulay module H * (BK 0 ), hence by [19,Lemma 5.4] (the graded version of [16,Lemma 4.3]…”

“…[13]. More recently, group actions whose equivariant cohomology satisfies this requirement have been investigated in [16], [19], and [18]. We adopt the terminology already used in those papers: if a group G acts on a space M in such a way that H * G (M ) is a Cohen-Macaulay H * (BG)-module, we simply say that the G-action is Cohen-Macaulay.…”

Equivariant cohomology, syzygies and orbit structure

“…In recent years, this "Atiyah-Bredon sequence" has been studied by Franz-Puppe [20], [22] and Goertsches-Töben [23]. Moreover, it is implicit in papers of De Concini-Procesi-Vergne on transversally elliptic operators, splines and the infinitesimal index [13], [14].…”

Equivariant cohomology of K-contact manifolds

“…See Definition 4.4 for the definition of equivariant formality of transverse actions. Using results from [13] and [14], these theorems will be deduced from the fact that a generic component of the contact moment map is a Morse-Bott function whose critical set equals the set of closed Reeb orbits.…”