Resolutions of moduli spaces and homological stability

Abstract

We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and give explicit stability ranges in many new cases. In each of these cases the stable homology can be identified using the methods of Galatius, Madsen, Tillmann and Weiss.

arXiv comment74 pages, 15 figures; V2: Warning, Completely Revised Version; V3: Removed discussion of configuration spaces; V4 Completely revised again, to appear in J. Eur. Math. Soc