Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2013; 50(4): 865-877

Printed July 1, 2013

https://doi.org/10.4134/JKMS.2013.50.4.865

Copyright © The Korean Mathematical Society.

The braidings in the mapping class groups of surfaces

Yongjin Song

Inha University

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 $\M$. 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 $\phi:B_g\ra\Gamma_{g,1}$. 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.

Keywords: braid group, mapping class group, Dehn twists, braided monoidal category, double loop space, plus construction

MSC numbers: 55R37, 18D10, 57M50, 55P48