From braid groups to mapping class groups

Abstract

In this paper, we classify homomorphisms from the braid group on n strands to the mapping class group of a genus g surface. In particular, we show that when \(g<n-2\), all representations are either cyclic or standard. Our result is sharp in the sense that when \(g=n-2\), a generalization of the hyperelliptic representation appears, which is not cyclic or standard. This gives a classification of surface bundles over the configuration space of the complex plane. As a corollary, we partially recover the result of Aramayona–Souto (Geom Topol 16(4):2285–2341, 2012), which classifies homomorphisms between mapping class groups, with a slight improvement.