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