The braidings in the mapping class groups of surfaces

TL;DR

This paper explores the structure of the mapping class groups of surfaces, revealing their braided monoidal nature, and establishes an injective, nongeometric map from braid groups to these surface groups, with implications for their homology.

Contribution

It provides a concrete geometric interpretation of the braiding in mapping class groups and constructs an injective, nongeometric embedding of braid groups into these groups.

Findings

The braiding in the mapping class groups has a clear geometric meaning.

An injective, nongeometric map from braid groups to mapping class groups is constructed.

The induced homology map is trivial in the stable range.

Abstract

The disjoint union of mapping class groups of surfaces forms a braided monoidal category $\mathcal M$, as the disjoint union of the braid groups $\mathcal B$ does. We give a concrete, and geometric meaning of the braiding $\beta_{r,s}$ in $\M$. Moreover, we find a set of elements in the mapping class groups which correspond to the standard generators of the braid groups. Using this, we obtain an obvious map $\phi:B_g\ra\Gamma_{g,1}$. We show that this map $\phi$ is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor $\Phi : \mathcal B \rightarrow \mathcal M$, the integral homology homomorphism induced by $\phi$ is trivial in the stable range.