Computations in $C_{pq}$-Bredon cohomology

TL;DR

This paper computes the $RO(C_{pq})$-graded Bredon cohomology of $C_{pq}$-orbits, revealing it depends on fixed point dimensions and generalizes earlier results for $C_p$, with applications to freeness theorems for certain spaces.

Contribution

It extends the computation of Bredon cohomology from $C_p$ to $C_{pq}$, providing new formulas and a freeness theorem for complex projective spaces and Grassmannians.

Findings

Cohomology groups depend on fixed point dimensions.

Generalization of Stong and Lewis results from $C_p$ to $C_{pq}$.

Freeness theorem applies to specific complex spaces.

Abstract

In this paper, we compute the $RO(C_{pq})$-graded cohomology of $C_{pq}$-orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group $C_p$. The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group $C_p$ was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem.