The braidings in the mapping class groups of surfaces
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The disjoint union of mapping class groups of surfaces formsa braided monoidal category M, as the disjoint union of the braid groups B does. We give a concrete and geometric meaning of the braidings βr,s in M. Moreover, we nd a set of elements in the mapping class groupswhich correspond to the standard generators of the braid groups. Usingthis, we can de ne an obvious map ø : Bg → Γg,1. We show that this map ø is injective and nongeometric in the sense of Wajnryb. Since thismap extends to a braided monoidal functor Ф : B → M, the integralhomology homomorphism induced by ø is trivial in the stable range.
Abstract
The disjoint union of mapping class groups of surfaces formsa braided monoidal category M, as the disjoint union of the braid groups B does. We give a concrete and geometric meaning of the braidings βr,s in M. Moreover, we nd a set of elements in the mapping class groupswhich correspond to the standard generators of the braid groups. Usingthis, we can de ne an obvious map ø : Bg → Γg,1. We show that this map ø is injective and nongeometric in the sense of Wajnryb. Since thismap extends to a braided monoidal functor Ф : B → M, the integralhomology homomorphism induced by ø is trivial in the stable range.
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