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Braids, mapping class groups, and categorical delooping

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Abstract

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map \(\phi : \beta_{2g} \to \Gamma _{g,1}\) from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257–175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, \(\phi\) is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology.

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Authors and Affiliations

  1. Department of Mathematics, Inha University, 253 Yonghyun-dong, Nam-gu, Incheon, 402-751, South Korea

    Yongjin Song

  2. Mathematical Institute, Oxford University, 24-29 St Giles Street, Oxford, OX1 3LB, UK

    Ulrike Tillmann

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  1. Yongjin Song
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  2. Ulrike Tillmann
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Correspondence to Ulrike Tillmann.

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The first author was supported by Inha University research grant.

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Song, Y., Tillmann, U. Braids, mapping class groups, and categorical delooping. Math. Ann. 339, 377–393 (2007). https://doi.org/10.1007/s00208-007-0117-z

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  • DOI: https://doi.org/10.1007/s00208-007-0117-z

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