Braids, mapping class groups, and categorical delooping

Details Braids, mapping class groups, and categorical delooping Braids, mapping class groups, and categorical delooping

2006-09-21

arxiv.org/abs/math/0609597v2

Mathematics

Algebraic Topology

19 pages, 9 figures, latex

Scientific paper

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with any trivial coefficients. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping. In an appendix we discuss geometrically defined homomorphisms from the braid to the mapping class group.

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