Homological monotonicity for configuration spaces of manifolds

Abstract

Consider the configuration spaces of manifolds. An influential theorem of McDuff, Segal and Church shows that the (co)homology of the unordered configuration space is independent of number of points in a range of degree called the stable range. We study the another important (and general) property of unordered configuration spaces of manifolds (not necessarily orientable, and not necessarily admitting non-vanishing vector field) that is homological monotonicity in unstable part. We show that the homological dimension of unordered configuration spaces of manifolds in each degree is monotonically increasing. Our results show that the monotonicity property is not depend on the differential structure and orientiability of manifold.