Homology of the family of hyperelliptic curves

TL;DR

This paper calculates the integral homology of a family of hyperelliptic curves, revealing they have only 2-torsion and exploring their stabilization properties and Poincaré series.

Contribution

It provides a complete computation of the homology of hyperelliptic curves with one boundary component, linking it to braid group actions and torsion properties.

Findings

Homology groups have only 2-torsion.

Explicit Poincaré series for stable and unstable homology.

Stability properties of the homology are analyzed.

Abstract

Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of the disk ramified over $n = 2g + 1$ points. The main part of such homology is described by the homology of the braid group with coefficients in a symplectic representation, namely the braid group $\mathrm{Br}_n$ acts on the first homology group of a genus $g$ surface via Dehn twists. Our computations shows that such groups have only $2$-torsion. We also investigate stabilization properties and provide Poincar\'e series, both for unstable and stable homology.