Basic Kirwan Surjectivity for K-Contact Manifolds

TL;DR

This paper extends Kirwan surjectivity to the setting of K-contact manifolds, establishing conditions under which the basic cohomology map is surjective and describing the kernel for certain group actions.

Contribution

It proves an analogue of Kirwan surjectivity for equivariant basic cohomology of K-contact manifolds and characterizes the kernel for specific group actions.

Findings

Reproduces Kirwan surjectivity for free $S^1$-actions on K-contact manifolds.

Provides a Tolman-Weitsman type description of the kernel of the basic Kirwan map.

Shows torus actions preserving the contact form are equivariantly formal in the basic setting.

Abstract

We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free $S^1$-action, the $S^1$-quotient is a symplectic manifold and our result reproduces Kirwan's surjectivity for these symplectic manifolds. We further prove a Tolman-Weitsman type description of the kernel of the basic Kirwan map for $S^1$-actions and show that torus actions on a K-contact manifold that preserve the contact form and admit 0 as a regular value of the contact moment map are equivariantly formal in the basic setting.