Title: Homology of the family of hyperelliptic curves,
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 Poincaré series, both for unstable and stable
homology.