Equivariant cohomology for cyclic groups of square-free order

The main objective of this paper is to compute RO(G)-graded cohomology of G-orbits for the group \(G=C_n\), where n is a product of distinct primes. We compute these groups for the constant Mackey functor \(\underline{\mathbb {Z}}\) and the Burnside ring Mackey functor \(\underline{A}\). Among other results, we show that the groups \(\underline{H}^\alpha _G(S^0)\) are mostly determined by the fixed point dimensions of the virtual representations \(\alpha \), except in the case of \(\underline{A}\) coefficients when the fixed point dimensions of \(\alpha \) have many zeros. In the case of \(\underline{\mathbb {Z}}\) coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain G-complexes.