Algebra/Topology Seminar

Speaker: Andrea Bianchi

Title: Polynomial stability of the homology of Hurwitz spaces

Abstract: This is joint work with Jeremy Miller. For a conjugation-invariant subset Q of a finite group G we consider the Hurwitz spaces Hur_n(Q), parametrising G-branched covers of the plane with exactly n branch points and local monodromies in Q. We are interested in stability phenomena of the homology groups H_i(Hur_n(Q)) for fixed i and increasing n. There are several stabilisation maps Hur_n(Q)--->Hur_{n+1}(Q), one for each element of Q; our main result is that for n large enough (compared to i), each class in H_i(Hur_{n+1}(Q)) is a combination of images of homology classes in H_i(Hur_n(Q)) along (possibly different) stabilisation maps. Taking coefficients in a field F, we show that the dimension of H_i(Hur_n(Q);F) agrees, for n large enough, with a quasi-polynomial function of n of controlled period and degree. Our work extends previous results of Ellenberg-Venkatesh-Westerland and rely on techniques introduced by them and by Hatcher-Wahl.