INVARIANT SULLIVAN–DE RHAM FORMS ON CYCLIC SETS

Abstract

Let Y be a topological space; and suppose that the circle group, S¹, which will be denoted by G throughout this paper, is acting on Y. Then the equivariant cohomology H*_G(Y) of Y is defined to be H*(Y_G), where Y_G=(EG×Y)/G, the Borel construction. Because of the bundle map Y_G→BG, the cohomology H*_G(Y) is an H*(BG)-module. With rational coefficients H*(BG; ℚ)=ℚ[u], where deg(u)=2. And, when Y is a reasonably nice finite-dimensional space, for example, a finite-dimensional G-CW-complex, then the localized module H*_G(Y; ℚ)[u⁻¹] is isomorphic to H*_G (Y^G; ℚ)[u⁻¹] via the restriction homomorphism, where Y^G⊆Y is the fixed point set. (See, for example, [2].) That this very useful Localization Theorem does not generalize to infinite dimensional spaces can be seen by taking Y=EG, for example.