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.
