Embeddings of braid groups into mapping class groups and their homology

TL;DR

This paper constructs various embeddings of braid groups into mapping class groups, revealing differences in their homological properties between orientable and non-orientable surfaces, and shows these embeddings are non-geometric.

Contribution

It introduces new families of embeddings of braid groups into mapping class groups and analyzes their homological effects, highlighting non-geometric embeddings.

Findings

Embeddings induce trivial stable homology in orientable cases.

Embeddings do not induce trivial stable homology in non-orientable cases.

Standard generators are not mapped to Dehn twists, indicating non-geometric embeddings.

Abstract

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.