Plant complexes and homological stability for Hurwitz spaces

TL;DR

This paper introduces plant complexes and demonstrates that homological stabilization of Hurwitz spaces depends solely on their zeroth homology groups, advancing understanding of their topological properties.

Contribution

It introduces plant complexes as a new class of simplicial complexes and generalizes a homological stabilization result for Hurwitz spaces.

Findings

Homological stabilization depends only on zeroth homology groups.

Plant complexes generalize arc complexes on surfaces.

Established new connections between topology and algebraic structures.

Abstract

We study Hurwitz spaces with regard to homological stabilization. By a Hurwitz space, we mean a moduli space of branched, not necessarily connected coverings of a disk with fixed structure group and number of branch points. We choose a sequence of subspaces of Hurwitz spaces which is suitable for our investigations. In the first part, we introduce and study plant complexes, a large new class of simplicial complexes, generalizing the arc complex on a surface with marked points. In the second part, we generalize a result by Ellenberg-Venkatesh-Westerland by showing that homological stabilization of our sequence of Hurwitz spaces depends only on properties of their zeroth homology groups.