Stability for closed surfaces in a background space

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 $\mathscr{S}_g(K)$. The homology stability of surfaces in $K$ with an arbitrary number of boundary components, $\mathscr{S}_{g,n}(K)$, was studied by the authors in a previous paper. 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 $\mathscr{S}_g(K)$, 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.