The braidings in the mapping class groups of surfaces

Yongjin Song

Published 2012-05-08Version 1

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.