The completion conjecture in equivariant cohomology

Consider RO (G)-graded an cohomology theory k G. We shall not insist on a detailed definition; suffice it to say that there is a suspension isomorphism for each real representation of G. The first examples were real and complex equivariant K-theory KO and K G. The next example was equivariant stable , cohomotopy theory ~G" There are RO(G)-graded ordinary cohomology theories with coefficients in Mackey functors. The study of these theories is still in its infancy. They can all be defined for arbitrary compact Lie groups, but we shall restrict our attention to finite groups. When we localize away from the order of G, there are very * powerful algebraic devices for the reduction of the calculation of k G to nonequivariant calculations. If we localize at a prime dividing the order of G, there are techniques for reducing calculations to consideration of p-groups contained in G. There are no known general procedures for the calculation of * k G at p for p-groups G. Largely for this reason, the reservoir of known calculations is almost empty. Let A(G) ring of finite G-sets. Then ~ takes values in be the Burnside the category of A(G)-modules. For some purposes, this is the ~ain reason for interest in the R0(G)-grading. The assertion is false for Z-graded equivariant cohomology theories which fail to extend to RO(G)-graded theories. Let EG be a free contractible G-CW complex and let e: EG ÷ * = {pt} be the trivial rap. We have an induced homomorphism of A(G)-modules