Stability in the homology of Deligne-Mumford compactifications

Abstract

Using the the theory of FS^op modules, we study the asymptotic behavior of the homology of ${\overline{M}}_{g,n}$, the Deligne--Mumford compactification of the moduli space of curves, for $n>>0$. An FS^op module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via maps that glue on marked P^1's, we give the homology of ${\overline{M}}_{g,n}$ the structure of an FS^op module and bound its degree of generation. As a consequence, we prove that the generating function ${\sum }_n\dim (H_i({\overline{M}}_{g,n}))t^n$ is rational, and its denominator has roots in the set $\{1,1/2,\dots ,1/p(g,i)\}$ where $p(g,i)$ is a polynomial of order $O(g^2{i}^2)$. We also obtain restrictions on the decomposition of the homology of ${\overline{M}}_{g,n}$ into irreducible $S_n$ representations.

arXiv commentCorrections to section 5, final version to appear in Compos. Math