Stability in the homology of Deligne-Mumford compactifications

TL;DR

This paper investigates the asymptotic homology behavior of Deligne-Mumford moduli spaces using FS^op modules, establishing rational generating functions and restrictions on symmetric group representations.

Contribution

It introduces an FS^op module framework for homology of ar M_{g,n} and proves rationality of generating functions with roots linked to polynomial bounds.

Findings

The homology generating function is rational.

Roots of the generating function denominator are in /2^k set.

Restrictions are found on the decomposition into irreducible S_n representations.

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.