Equivariant cohomology of K-contact manifolds

Details Equivariant cohomology of K-contact manifolds Equivariant cohomology of K-contact manifolds

2011-02-22

arxiv.org/abs/1102.4476v2

Mathematics

Symplectic Geometry

22 pages. v2 corrects minor inaccuracies

Scientific paper

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, which is a generalization of equivariant formality for torus actions without fixed points. As a consequence, a generic component of the contact moment map is a perfect Morse-Bott function 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 is trivial in odd degrees, and its dimension equals the number of closed Reeb orbits. We characterize the 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 us to calculate the equivariant cohomology algebra of K-contact manifolds in presence of the nonisolated GKM condition.

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