The equivariant thom isomorphism theorem

Abstract

In this paper we extend ordinary RO(G)-graded cohomology to a theory graded on virtual G-bundles over a G-space and show that a Thorn Isomorphism theorem for general G-vector bundles results. Our approach uses Elmendorf s topologized spectra. We also show that the grading can be reduced from the group of virtual G-vector bundles over a space to a quotient group, using ideas from a new theory of equivariant orientations. As an application of the Thorn Isomorphism theorem, we give a new calculation of the additive structure of the equivariant cohomology of complex projective spaces for G-Z/p, partly duplicating and partly extending a recent calculation done by Lewis. © 1992 by Pacific Journal of Mathematics.