Equivariant cohomology and analytic descriptions of ring isomorphisms (Q1006836)

A class of connected closed \(G\)-manifolds with a nonempty finite fixed point set is considered in the case \(G=\mathbb Z_2\). Using the equivariant index, an explicit description of the equivariant cohomology of such a \(G\)-manifold is given in terms of algebra. Thus, analytic descriptions of ring isomorphisms among equivariant cohomology rings of such \(G\)-manifolds, and a necessary and sufficient condition that the equivariant cohomology rings of such two \(G\)-manifolds are isomorphic, is obtained. This also leads the authors to analyze how many equivariant cohomology rings up to isomorphism there are for such \(G\)-manifolds in the 2- and 3-dimensional case.