Hamiltonian S1-spaces of dimension four are classified by the associated decorated graphs. We give a generators and relations description of the equivariant cohomology of a Hamiltonian S1-space. We use the description to show that the equivariant cohomology determines the decorated graph, sans the heights and area labels, up to flipping chains. As a result, we get a proof of the finiteness of Hamiltonian circle actions on a closed symplectic four-manifold, that does not use pseudo-holomorphic tools.

This is a joint work with Tara Holm.