From braid groups to mapping class groups

Lei Chen, Aru Mukherjea

Published 2020-11-25Version 1

In this paper, we classify homomorphisms from the braid group of $n$ 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. This gives a classification of genus $g$ surface bundles over the configuration space of the complex plane. As a corollary, we recover partially the result of Aramayona-Souto with a slight improvement, which classifies homomorphisms between mapping class groups.