Stability in the homology of Deligne-Mumford compactifications

Using the the theory of FS^op modules, we study the asymptotic behavior of the homology of ${\bar{M}}_{g,n}$, 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 ${\bar{M}}_{g,n}$ the structure of an FS^op module and bound its degree of generation. As a consequence, we prove that the generating function $\sum_{n} \dim(H_i(\overline M_{g,n})) t^n$isrational,anditsdenominatorhasrootsintheset${1, 1/2, \dots, 1/p(g,i)}$where$p(g,i)$isapolynomialoforder$O(g^2 i^2)$. We also obtain restrictions on the decomposition of the homology of $\overline M_{g,n}$intoirreducible$S_n$ representations.