Ring structures of rational equivariant cohomology rings and ring homomorphisms between them (Q2839881)

This paper analyzes the \(S^1\)-equivariant cohomology ring of a closed oriented manifold \(M\) endowed with an \(S^1\)-action with finite non-empty fixed point set. The manifold \(M\) is assumed to be \(S^1\)-equivariantly formal, meaning that the inclusion of \(M\) as a typical fiber in the Borel fibration \(M_{S^1}:=ES^1\times_{S^1} M\to BS^1\) is surjective in cohomology (the equivariant cohomology of \(M\) is the cohomology of \(M_{S^1}\) with \(\mathbb Q\) coefficients). Such manifolds only exist in even dimensions. The sum of the Betti numbers of \(M\) equals the number of fixed points. The authors prove that the equivariant cohomology rings of two manifolds as above of dimension \(2n\in\{2,4\}\) are isomorphic if and only the manifolds have the same number of fixed points. For arbitrary \(n\), they show that the direct implication still holds for maps in cohomology induced by \(S^1\)-equivariant maps.