THE EQUIVARIANT THOM ISOMORPHISM THEOREM

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 Thom 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 Thom 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.