Algebraic structures on cohomology of configuration spaces of manifolds with flows

2015/08/10 by Ellenberg, Jordan S., Wiltshire-Gordon, John D.
#18D50 #20C15 #55R80 #57R19 #57R40 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1508.02430

Abstract

Let PConfⁿ M be the configuration space of ordered n-tuples of distinct points on a smooth manifold M admitting a nowhere-vanishing vector field. We show that the ith cohomology group with coefficients in a field Hⁱ(PConfⁿ M, k) is an N-module, where N is the category of noncommutative finite sets introduced by Pirashvili and Richter. Studying the representation theory of N, we obtain new polynomiality results for the cohomology groups Hⁱ(PConfⁿ M, k). In the case of unordered configuration space Confⁿ M = (PConfⁿ M)/S_n and rational coefficients, we show that cohomology dimension in fixed degree is nondecreasing.

Related

• Computing cohomology of configuration spaces
• Configuration spaces in algebraic topology
• The Cohomology Algebra of Unordered Configuration Spaces
• Arithmeticity for integral cohomological dimension of configuration spaces of manifolds
• The cohomology algebra of unordered configuration spaces