The braidings in the mapping class groups of surfaces

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 mapp...

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Bibliographic Details
Published in:arXiv.org 2012-05
Main Author: Song, Yongjin
Format: Article
Language:English
Subjects:
Braid theory
Braiding
Homology
Homomorphisms
Mapping
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Summary: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.
ISSN:2331-8422