Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II

TL;DR

This paper advances the understanding of the Cohen-Lenstra conjecture over function fields by establishing homological stability results for Hurwitz spaces, linking topology and arithmetic to confirm conjectural distributions.

Contribution

It completes previous work by proving a version of the Cohen-Lenstra conjecture over function fields, using new topological and arithmetic theorems involving homology limits and Galois actions.

Findings

Computed the direct limit of homology over puncture-stabilization of mapping spaces

Determined the Galois action on stable components of Hurwitz schemes

Established homological stability results connecting topology and arithmetic

Abstract

We prove a version of the Cohen--Lenstra conjecture over function fields (completing the results of our prior paper). This is deduced from two more general theorems, one topological, one arithmetic: We compute the direct limit of homology, over puncture-stabilization, of spaces of maps from a punctured manifold to a fixed target; and we compute the Galois action on the set of stable components of Hurwitz schemes.