Homology, Homotopy and Applications

Volume 13 (2011)

Number 2

Stability for closed surfaces in a background space

Pages: 301 – 313

DOI: https://dx.doi.org/10.4310/HHA.2011.v13.n2.a18


Ralph L. Cohen (Department of Mathematics, Stanford University, Stanford, Calif., U.S.A.)

Ib Madsen (Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark)


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.

2010 Mathematics Subject Classification

30F99, 57M07, 57R50

Published 25 January 2012