From braid groups to mapping class groups

TL;DR

This paper classifies homomorphisms from braid groups to mapping class groups of surfaces, revealing that most are cyclic or standard except in a specific case where a hyperelliptic representation appears, advancing understanding of surface bundles.

Contribution

It provides a sharp classification of braid group homomorphisms into mapping class groups, including a new hyperelliptic case when genus equals n-2.

Findings

Homomorphisms are cyclic or standard when g < n-2

A hyperelliptic representation appears at g = n-2

Partial recovery and improvement of Aramayona-Souto's classification

Abstract

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, 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, which classifies homomorphisms between mapping class groups, with a slight improvement.