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

2024/01/09 by Dugger, Daniel, Hazel, Christy
#55N91 #55P91 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.04826

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)≥ 3 and G is any of C_p, C_p² (p a prime), or the symmetric group on three letters.

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