Celebratio Mathematica — Birman

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Works connected to Tara Elise Brendle

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J. S. Birman and T. E. Brendle: “Braids: A survey,” Chapter 2, pp. 19–103 in Handbook of knot theory. Edited by W. Menasco and M. Thistlethwaite. Elsevier B. V. (Amsterdam), 2005. MR 2179260 Zbl 1094.57006 incollection

This article is about Artin’s braid group \( \mathbf{B}_n \) and its role in know theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid group generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labeled, as we encounter them. A guide to computer software is given together with an extensive bibliography.

Morwen Bernard Thistlethwaite	Related
William Wyatt Menasco	Related
Tara Elise Brendle	Related

J. S. Birman, T. E. Brendle, and N. Broaddus: “Calculating the image of the second Johnson–Morita representation,” pp. 119–134 in Groups of diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday (Tokyo, 11–15 September 2006). Edited by R. C. Penner. Advanced Studies in Pure Mathematics 52. Mathematical Society of Japan (Tokyo), 2008. MR 2509709 Zbl 1183.57016 ArXiv 0708.3861 incollection

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus \( g \) with one boundary component to \( \bigwedge^3H \), the third exterior product of the homology of the surface. Morita then extended Johnson’s homomorphism to a homomorphism from the entire mapping class group to \[ \tfrac{1}{2} \bigwedge^3 H \rtimes \operatorname{Sp}(H) .\] This Johnson–Morita homomorphism is not surjective, but its image is finite index in \( \frac{1}{2} \) \( \bigwedge^3 H \rtimes \) \( \operatorname{Sp}(H) \) [Morita 1993]. Here we give a description of the exact image of Morita’s homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.

Tara Elise Brendle	Related
Nathan Darrell Broaddus	Related
Robert Clark Penner	Related
Shigeyuki Morita	Related