$RO(G)$-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.

Figures (6)

• Figure 1: The rings $H^\star(S(2\sigma);\underline{{\mathbb Z}})$ and $H^\star(S(3\sigma);\underline{{\mathbb Z}})$
• Figure 2: The additive structure of $H^\star(\mathop{\mathrm{OC}}\nolimits_3(2\sigma);\underline{{\mathbb Z}})$
• Figure 3: The module $\mathbb M_{ev}$
• Figure 4: The module $\mathbb M_{odd}$
• Figure 5: The module $\mathbb M_{0}$

Theorems & Definitions (80)

• Remark 1.1
• Remark 2.2
• Proposition 2.4: Quotient Lemma
• Proposition 2.5
• proof
• Proposition 2.7
• proof : Proof of Proposition \ref{['pr:a-group']}
• Remark 2.8
• Proposition 2.9
• proof