Computations in the unstable homology of moduli spaces of Riemann surfaces

TL;DR

This paper surveys homology computations for moduli spaces of Riemann surfaces, highlighting the complexities of unstable homology and providing explicit generators using various homology operations.

Contribution

It offers a comprehensive overview of homology calculations in unstable regimes and determines explicit generators with integral, mod-2, and rational coefficients.

Findings

Homology computations for moduli spaces with various coefficients.

Explicit generators identified using homology operations.

Unstable homology remains complex despite stable results.

Abstract

In this article we give a survey of homology computations for moduli spaces $\mathfrak{M}_{g,1}^m$ of Riemann surfaces with genus $g\geqslant 0$, one boundary curve, and $m\geqslant 0$ punctures. While rationally and stably this question has a satisfying answer by the Madsen-Weiss theorem, the unstable homology remains notoriously complicated. We discuss calculations with integral, mod-2, and rational coefficients. Furthermore, we determine, in most cases, explicit generators using homology operations.