Equivariant cohomology for cyclic groups of square-free order

TL;DR

This paper computes the RO(G)-graded cohomology of G-orbits for cyclic groups of square-free order, revealing how fixed point dimensions influence the cohomology groups and their ring structures, with applications to freeness of G-complexes.

Contribution

It provides explicit calculations of RO(G)-graded cohomology for cyclic groups of square-free order with different Mackey functor coefficients, advancing understanding of equivariant cohomology structures.

Findings

Cohomology groups are mainly determined by fixed point dimensions of representations.

The ring structure on cohomology with constant integer coefficients is described.

Freeness results for certain G-complexes are established.

Abstract

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 $\underline{Z}$ and for the Burnside ring Mackey functor $\underline{A}$. Among other things, 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 $\underline{Z}$ coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain $G$-complexes.