Plant complexes and homological stability for Hurwitz spaces

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 (Sections 2 to 4), 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 (Sections 5 to 7), we generalize a result from [EVW16] by showing that homological stabilization of our sequence of Hurwitz spaces depends only on properties of their zeroth homology groups. Contents1. Introduction 1 2. Plants and plant complexes 5 3. Connectivity analysis 8 4. Combinatorics of colored plant complexes 13 5. Hurwitz spaces 21 6. Homological stability for Hurwitz spaces 23 7. Application 30 References 33