On homological stability for configuration spaces on closed background manifolds
with Federico Cantero

Documenta Mathematica vol. 20 (2015) pp. 753-805
arXiv: abs — pdf
v4 — v3 — v2 — v1

Abstract

We introduce a new map between configuration spaces of points in a background manifold — the replication map — and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the background manifold is closed, in which case the classical stabilisation map does not exist. We then establish conditions on the manifold and on the coefficients under which homological stability holds for configuration spaces on closed manifolds. These conditions are sharp when the background manifold is a two-dimensional sphere, the classical counterexample in the field. For field coefficients this extends results of [Church] and [Randal-Williams] to the case of odd characteristic, and for p-local coefficients it improves results of [Bendersky and Miller].

• [Bendersky and Miller] M. Bendersky and J. Miller. (2014) Localization and homological stability of configuration spaces. Q. J. Math., 65.3, pp. 807–815.
• [Church] T. Church. (2012) Homological stability for configuration spaces of manifolds. Invent. Math., 188.2, pp. 465–504.
• [Randal-Williams] O. Randal-Williams. (2013) Homological stability for unordered configuration spaces. Q. J. Math., 64.1, pp. 303–326.