On the equivariant cohomological rigidity of semi-free Hamiltonian circle actions

Abstract

We consider semi-free Hamiltonian $S^1$-manifolds of dimension six and establish when the equivariant cohomology and data on the fixed point set determine the isomorphism type. Gonzales listed conditions under which the isomorphism type of such spaces is determined by fixed point data. We pointed out in an earlier paper that this result as stated is erroneous, and proved a corrected version. However, that version relied on a certain distribution of fixed points that is not at all necessary. In this paper, we replace the latter assumption with a global assumption on equivariant cohomology that is necessary for an isomorphism. We also extend our result to the equivariant (non-symplectic) topological category. The variation in the earlier paper was tailored to suit the requirements of Cho's application of Gonzales' statement to classify semi-free monotone, Hamiltonian $S^1$-manifolds of dimension six. In the current paper, we aim to give the definitive statement relating fixed point data and equivariant cohomology to the isomorphism type of a semi-free Hamiltonian $S^1$-manifold.