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 Sg(K). The homology stability of surfaces in K with an arbitrary number of boundary components, Sg,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, Γg,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 Sg(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.
Homology, Homotopy and Applications, Vol. 13 (2011), No. 2, pp.301-313.
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