Representation Stability, Configurations Spaces, and Deligne-Mumford Compactifications

Abstract

We study the homology of ordered configuration spaces and Deligne--Mumford compactifications using tools from representation stability.

In the case of configuration spaces, we show that for topological spaces that are greater than or equal to 2 dimensional in some sense, the cohomology of configuration space is a finitely generated module over the category of finite sets and injections. From this, we deduce homological stability results for unordered configuration space.

In the case of Deligne--Mumford compactifications, we show that the homology is a finitely generated module over the opposite of the category of finite sets and surjections. As a consequence, we show that the generating function of the homology of these spaces is rational and takes a specific form. In particular, the dimension of the homology groups eventually agrees with a sum of polynomials times natural number exponentials.