From braid groups to mapping class groups

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. 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 [AS12] with a slight improvement, which classifies homomorphisms between mapping class groups.

The generalized Harer conjecture for the homology triviality

“…Those interchanged subparts may be points or subsurfaces, not necessarily connected. Although the classification of embeddings from braid groups to mapping class groups is still widely open (e.g., [5]), all known embeddings are of this kind. Note that a standard generator of braid group is, in principle, characterized as interchanging two points by a half twist.…”

Homomorphisms of commutator subgroups of braid groups

“…(3) Caplinger and the first author [9] show that the smallest non-abelian finite quotients of 𝐵 5 and 𝐵 6 are the corresponding symmetric groups. (4) Chen and Mukherjea [11] classify homomorphisms from 𝐵 𝑛 to the mapping class group of a surface of genus g ⩽ 𝑛 − 3. (5) Scherich and Verberne [21] improved on the aforementioned lower bound of Chudnovsky, Li, Partin, and the first author.…”

On virtual embeddings of braid groups into mapping class groups of surfaces

“…Recently, Chen-Mukherjea in [15] proved that any non-trivial homomorphism from a braid group to the pure mapping class groups of a surface with punctures is a transvection of a geometric monodromy if the genus of the surface lies in a certain range.…”