Equivariant cohomology does not distinguish Hamiltonian S^1-manifolds

For Hamiltonian circle actions on 4-manifolds, we give an example of an isomorphism of equivariant cohomology modules that cannot be induced by an equivariant diffeomorphism of spaces. This is in contrast to Masuda's result establishing that in the toric case, the equivariant cohomology module determines the manifold. We also give a soft proof that there are finitely many maximal Hamiltonian circle actions on a fixed closed symplectic 4-manifold. We achieve these results by giving a generators and relations description in terms of the decorated graph for the even part of the equivariant cohomology as a module over the equivariant cohomology of a point. 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 modules are weakly isomorphic (and the odd groups have the same ranks) if and only if the labelled graphs obtained from the decorated graphs by forgetting the height and area labels are isomorphic.