# THE BRAIDINGS IN THE MAPPING CLASS GROUPS OF SURFACES

• Song, Yongjin (Departments of Mathematics, Inha University)
• Published : 2013.07.01
• 54 6

#### 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 braidings ${\beta}_{r,s}$ in $\mathcal{M}$. 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 can define an obvious map ${\phi}\;:\;B_g{\rightarrow}{\Gamma}_{g,1}$. 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.

#### Keywords

braid group;mapping class group;Dehn twists;braided monoidal category;double loop space;plus construction

#### Acknowledgement

Supported by : National Foundation of Korea (NRF)

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