Stability in the homology of configuration spaces of high dimensional manifolds

Abstract

Consider the configuration spaces of manifold (closed or open). An influential theorem of McDuff and Segal shows that the (co)homology of unordered configuration spaces of open manifold is independent of number of configuration points in a range of degree called the stable range. Church prove that the homological stability is also present in the closed background manifold. The Church's stable range is only depend on the number of configuration points. In this paper, we prove that if the cohomology of manifold (not necessarily orientable) is concentrated in even degree than the stable range for odd degree homology groups of configuration spaces is also depend on the dimension of background manifold. Moreover, we prove that if the second cohomology group of manifold is nontrivial then our stable range for even degree homology groups is optimal, and this stable range is depend only on the number of configuration points.