Stability for closed surfaces in a background space

TL;DR

This paper provides a new homotopy-theoretic proof of the rational homological stability of the moduli space of closed surfaces in a simply connected background space, extending previous results and including stability for surfaces with marked points.

Contribution

It introduces a novel proof technique for homological stability of closed surface moduli spaces in a background space, broadening understanding beyond boundary component cases.

Findings

Homological stability for closed surfaces in K proven using new methods

Stability results extended to surfaces with marked points

Provides a homotopy theoretic proof approach

Abstract

In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space $K$, which we denote by $S_g (K)$. The homology stability of surfaces in $K$ with an arbitrary number of boundary components, $S_{g,n} (K)$ was studied by the authors in \cite{cohenmadsen}. The study there relied on stability results for the homology of mapping class groups, $\Gamma_{g,n}$ with certain families of twisted coefficients. It turns out that these mapping class groups only have homological stability when $n$, the number of boundary components, is positive, or in the closed case when the coefficient modules are trivial. Because of this we present a new proof of the rational homological stability for $S_g(K)$, that is homotopy theoretic in nature. We also take the opportunity to prove a new stability theorem for closed surfaces in $K$ that have marked points.