From braid groups to mapping class groups

Lei Chen; Aru Mukherjea

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arXiv · 2011.13020

From braid groups to mapping class groups

Lei Chen·Aru Mukherjea

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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.

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Lei Chen, Aru Mukherjea. 2020-11-25. From braid groups to mapping class groups. https://arxiv.org/abs/2011.13020

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