Algebraic structures on cohomology of configuration spaces of manifolds with flows

Abstract

Let PConf^n 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 H^i(PConf^n 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 H^i(PConf^n M, k). In the case of unordered configuration space Conf^n M = (PConf^n M)/S_n and rational coefficients, we show that cohomology dimension in fixed degree is nondecreasing.