Equivariant cohomological rigidity for four-dimensional Hamiltonian $\mathbf{S^1}$-manifolds

TL;DR

This paper proves that for four-dimensional Hamiltonian circle actions on compact symplectic manifolds, the equivariant cohomology ring completely determines the equivariant diffeomorphism type, with isomorphisms arising from actual diffeomorphisms.

Contribution

It establishes that equivariant cohomology rings fully classify four-dimensional Hamiltonian circle actions up to equivariant diffeomorphism, strengthening previous results.

Findings

Equivariant cohomology rings determine the diffeomorphism type.

Isomorphisms of cohomology rings are induced by diffeomorphisms.

Classification of Hamiltonian circle actions in four dimensions.

Abstract

For manifolds equipped with group actions, we have the following natural question: To what extent does the equivariant cohomology determine the equivariant diffeotype? We resolve this question for Hamiltonian circle actions on compact, connected symplectic four-manifolds. They are equivariantly diffeomorphic if and only if their equivariant cohomology rings are isomorphic as algebras over the equivariant cohomology of a point. In fact, we prove a stronger claim: each isomorphism between their equivariant cohomology rings is induced by an equivariant diffeomorphism.