Equivariant cohomology of a complexity-one four-manifold is determined by combinatorial data

For Hamiltonian circle actions on compact, connected four-dimensional manifolds, we give a generators and relations description for the even part of the equivariant cohomology, as an algebra over the equivariant cohomology of a point. This description depends on combinatorial data encoded in the decorated graph of the manifold. We then give an explicit combinatorial description of all weak algebra isomorphisms. We use this description to prove that the even parts of the equivariant cohomology algebras are weakly isomorphic and the odd groups have the same ranks if and only if the labeled graphs obtained from the decorated graphs by forgetting the height and area labels are isomorphic. As a consequence, we give an example of an isomorphism of equivariant cohomology algebras that cannot be induced by an equivariant diffeomorphism of manifolds preserving a compatible almost complex structure. We also provide a soft proof that there are finitely many maximal Hamiltonian circle actions on a fixed compact, connected, four-dimensional symplectic manifold.