The equivariant cohomology of Hamiltonian \(G\)-spaces from residual \(S^1\) actions (Q5945787)

The authors show that for a Hamiltonian action of a compact torus \(G\) on a compact connected symplectic manifold \(M\), the \(G\)-equivariant cohomology is determined by the residual \(S^1\)-action on the submanifolds of \(M\) fixed by codimension-1 tori. This theorem allows to compute the equivariant cohomology of certain manifolds, which have pieces that are four-dimensional or smaller. As example, they compute the \(S^1\)-equivariant cohomology of \(CP^2\) with a Hamiltonian circle action, the \(T^2\)-equivariant cohomology of \(CP^3\) and the \(S^1\)-equivariant cohomology of a 4-dimensional manifold.