A classical theorem of McDuff and Segal states that the sequence of unordered configuration spaces Cn(M) associated to a connected, open manifold M satisfies a phenomenon called homological stability. This means that in each fixed degree q, the sequence of homology groups Hq(C_n(M)) is eventually constant. On the other hand, it is well-known that this fails for closed manifolds -- although some conditional results are known if one takes homology with coefficients in a more general ring than the integers.
In this talk, I will explain some recent joint work with Federico Cantero, in which we extend the previously known results in this situation. A key idea in our proof is to introduce so-called "replication maps" between configuration spaces, and show that these induce isomorphisms on homology in a range of degrees under certain conditions.
One corollary of our results is to recover a "homological periodicity" theorem of Nagpal -- if we take homology with field coefficients, then for each fixed q the sequence Hq(Cn(M)) is eventually periodic in n -- and obtain a much more explicit estimate for the period. Another corollary is that for odd-dimensional manifolds M, the two sequences C{2n}(M) and C{2n+1}(M) are (independently) homologically stable, even for integral coefficients.