From braid groups to mapping class groups
Lei Chen, Aru Mukherjea

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 strands to the mapping class group of a genus surface. In particular, we show that when , all representations are either cyclic or standard. Our result is sharp in the sense that when , 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory
