Celebratio Mathematica — Birman

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Works connected to Nathan Darrell Broaddus

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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

J. Birman, N. Broaddus, and W. Menasco: “Finite rigid sets and homologically nontrivial spheres in the curve complex of a surface,” J. Topol. Anal. 7 : 1 (2015), pp. 47–71. MR 3284389 Zbl 1308.57009 article

Aramayona and Leininger have provided a “finite rigid subset” \( \mathfrak{X}(\Sigma) \) of the curve complex \( \mathscr{C}(\Sigma) \) of a surface \( \Sigma = \Sigma_g^n \), characterized by the fact that any simplicial injection \[ \mathfrak{X}(\Sigma) \to \mathscr{C}(\Sigma) \] is induced by a unique element of the mapping class group \( \operatorname{Mod}(\Sigma) \). In this paper we prove that, in the case of the sphere with \( n \geq 5 \) marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a \( \operatorname{Mod}(\Sigma) \)-module generator for the reduced homology of the curve complex \( \mathscr{C}(\Sigma) \), answering in the affirmative a question posed in [Aramayona and Leininger 2013]. For the surface \( \Sigma = \Sigma_g^n \) with \( g \geq 3 \) and \( n \in \{0,1\} \) we find that the finite rigid set \( \mathfrak{X}(\Sigma) \) of Aramayona and Leininger contains a proper subcomplex \( X(\Sigma) \) whose reduced homology class is a \( \operatorname{Mod}(\Sigma) \)-module generator for the reduced homology of \( \mathscr{C}{\Sigma} \) but which is not itself rigid.

Nathan Darrell Broaddus	Related
William Wyatt Menasco	Related