The equivariant cohomology of complexity one spaces

Complexity one spaces are an important class of examples in symplectic geometry. They are less restrictive than toric symplectic manifolds. Delzant has established that toric symplectic manifolds are completely determined by their moment polytope. Danilov proved that the ordinary and equivariant cohomology rings are dictated by the combinatorics of this polytope. These results are not true for complexity one spaces. In this paper, we describe the equivariant cohomology for a Hamiltonian S^1\circlearrowright M^4 . We then assemble the equivariant cohomology of a complexity one space from the equivariant cohomology of the 2- and 4 -dimensional pieces, as a subring of the equivariant cohomology of its fixed points. We also show how to compute equivariant characteristic classes in dimension four.