$RO(G)$-graded Bredon cohomology of Euclidean configuration spaces

Abstract

Let $G$ be a finite group and $V$ be a $G$-representation. We investigate the $RO(G)$-graded Bredon cohomology with constant integral coefficients of the space of ordered configurations in $V$. In the case that $V$ contains a trivial subrepresentation, we show the cohomology is free as a module over the cohomology of a point, and we give a generators-and-relations description of the ring structure. In the case that $V$ does not contain a trivial representation, we give a computation of the module structure that works as long as a certain vanishing condition holds in the Bredon cohomology of a point. We verify this vanishing condition holds in the case that $\dim(V)\geq 3$ and $G$ is any of $C_p$, $C_{p^2}$ ($p$ a prime), or the symmetric group on three letters.