Equivariant cohomology of K-contact manifolds

Abstract

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation F of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of F vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.2010 Mathematics Subject Classification. 53D35, 53D20, 55N25.

On the topology of metric f–K-contact manifolds

“…, ξ s on a metric f -K-contact manifold M to the basic cohomology H * (M, F). We generalizes results of [16] in the K-contact case. The main tool is the torus T given by the closure of the flows of the characteristic vector fields in the isometry group of M , and a T -invariant Morse-Bott function S whose critical set C is equal to the union of closed leaves of F. This function generalizes a generic component of the contact momentum map in the K-contact setting, see [24,Section 4].…”

“…In this section we relate the ordinary and basic cohomology of a compact metric f -K-contact manifold (M 2n+s , f, ξ α , η α , g) to the union C of the closed leaves of the characteristic foliation F. This generalizes results from [24] and [16].…”

“…For s = 1, i.e., in the K-contact setting, the following theorem was known previously -the statement about the minimal number of closed leaves generalizes [24,Corollary 1], and the equivalence of the four conditions results from [16]. Theorem 6.4.…”

“…Example 6.6. It was shown in [16] that there is a K-contact structure with exactly four closed Reeb orbits on the 7-dimensional Stiefel manifold SO(5)/SO(3), which is a real cohomology sphere. All (iterated) mapping tori of automorphisms of this example thus satisfy the conditions in Theorem 6.4.…”

Equivariant de Rham cohomology: theory and applications

“…In He's paper an important feature of the class of actions he considers is that the one-skeleton of the action is the union of (three-dimensional) submanifolds each containing an arbitrary number of fixed point components, contrary to the the classical case in which the invariant two-spheres always contain exactly two fixed points. GKM theory for actions without fixed points was considered in [42], for a certain class of Cohen-Macaulay torus actions (see Section 12 below). Instead of the one-skeleton of the action one describes the equivariant cohomology of the action in terms of the b + 1-skeleton M b+1 of the action, where b is the lowest occurring dimension of an orbit.…”