Unordered configuration spaces on (connected) manifolds are basic objects

that appear in connection with many different areas of topology. When the

manifold M is non-compact, a theorem of McDuff and Segal states that these

spaces satisfy a phenomenon known as homological stability: fixing q, the

homology groups H_q(C_k(M)) are eventually independent of k. Here, C_k(M)

denotes the space of k-point configurations and homology is taken with

coefficients in Z. However, this statement is in general false for closed

manifolds M, although some conditional results in this direction are known.

I will explain some recent joint work with Federico Cantero, in which we

extend all the previously known results in this situation. One key idea is

to introduce so-called "replication maps" between configuration spaces,

which in a sense replace the "stabilisation maps" that exist only in the

case of non-compact manifolds. One corollary of our results is to recover a

"homological periodicity" theorem of Nagpal -- taking homology with field

coefficients and fixing q, the sequence of homology groups H_q(C_k(M)) is

eventually periodic in k -- and we obtain a much simpler estimate for the

period. Another result is that homological stability holds with Z[1/2]

coefficients whenever M is odd-dimensional, and in fact we improve this to

stability with Z coefficients for 3- and 7-dimensional manifolds.

20 April 2015

15:45

Martin Palmer

Abstract