Arithmeticity for integral cohomological dimension of configuration spaces of manifolds

Abstract

Consider the configuration spaces of manifolds. We give a precise formula for the integral cohomological dimension (the degree of top non-trivial integral cohomology group) of unordered configuration spaces of manifolds with non-trivial co-dimension one cohomology group, and show that the the sequence of cohomological dimensions is arithmetic. This arithmeticity is not present in the classical example of Arnold. Moreover, we show that the top integral cohomology group is infinite. Furthermore, We give a lower bound for the rank of top integral cohomology group. We also predict that the top integral cohomology group of configuration spaces of manifolds with non-trivial co-dimension one cohomology group is eventually finite. To the best of our knowledge, there is no rigorous bound for the cohomological dimension of ordered configuration spaces. As an application of main results, we give a sharp lower bound for the cohomological dimension of ordered configuration spaces of manifolds. The first step of the proof of main results is to define a reduced Chevalley Eilenberg complex.