The braidings in the mapping class groups of surfaces

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 $ϕ:B_g\raΓ_{g,1}$. We show that this map $\varphi$ is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor $\Phi :\mathcal{B}\to \mathcal{M}$, the integral homology homomorphism induced by $\varphi$ is trivial in the stable range.

11pages, 10 figures

References in corpus (1)

• Classifying spaces for braided monoidal categories and lax diagrams of bicategories