Homological stability for the moduli space of Riemann surfaces with boundary

Abstract

Moduli spaces frequently arise as solutions to classification problems. Showing that a collection of objects can be given the structure of a geometric space facilitates the research of a possible parametrization on the resulting space. The goal of the present work is to prove homological stability results for the moduli space of Riemann surfaces, exclusively looking at the case of Riemann surfaces with non-empty boundary and fixed positive length at the boundary components. The reason why we restrict to such a condition can be found in the famous and often quoted article "Stability of the Homology of the Mapping Class Groups of Orientable Surfaces" published in the Annals of Mathematics by John Harer in 1985. The stability results he addresses refer to the homology of an algebraic invariant for manifolds, the so-called mapping class group. As a matter of fact, it turns out that, given a connected, compact and oriented surfaces $S$ with non-empty boundary, the moduli space of Riemann surfaces $\operatorname{M}(S)$ is a model for the classifying space of the mapping class group $\operatorname{MCG}(S)$. Therefore, under this condition, the homology of the mapping class groups of $S$ equals the homology of $\operatorname{M}(S)$. The relevance of the Harer's stability theorem lies in its application in the Mumford's conjecture, formulated in 1983 by David Mumford and solved by Ib Madsen and Michael Weiss in 2007. According to the Mumford conjecture, the rational cohomology ring of the moduli space of Riemann surfaces is a polynomial algebra on the so-called Mumford-Morita-Miller classes, in a range of degrees increasing with the genus of the surface. Despite the numerous improvements of Harer's result, due to professor Nikolai V. Ivanov in 1989, Harer itself in 1993, O. Randal-Williams in 2009, and many more over a range of 35 years, variations of the paper are subjects of nowadays active research: the last attempt to improve the statement was achieved by Søren K. Boldsen in 2010 in the paper "Improved homological stability for the mapping class group with integral or twisted coefficient"; professor Nathalie Wahl reorganized all the literature concerning the stability of the mapping class groups in her paper “Homological stability for mapping class group of surfaces”, published in the third volume of the "Handbook of the moduli" in 2013