Resolutions of moduli spaces and homological stability

TL;DR

This paper introduces new semi-simplicial resolutions for moduli spaces of surfaces with tangential structures, leading to improved homological stability results and explicit stability ranges, enhancing understanding of their stable homology.

Contribution

It develops partial semi-simplicial resolutions for moduli spaces, resulting in stronger homological stability theorems and explicit stability ranges in many cases.

Findings

Improved homological stability ranges for moduli spaces of surfaces.

Explicit stability ranges identified in numerous new cases.

Stable homology characterized using methods of Galatius, Madsen, Tillmann, and Weiss.

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.