Calculating the image of the second Johnson-Morita representation. (English) Zbl 1183.57016

Birman, Joan S.; Brendle, Tara E.; Broaddus, Nathan

Penner, Robert (ed.) et al., Groups of diffeomorphisms in honor of Shigeyuki Morita on the occasion of his 60th birthday. Based on the international symposium on groups and diffeomorphisms 2006, Tokyo, Japan, September 11–15, 2006. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-48-8/hbk). Advanced Studies in Pure Mathematics 52, 119-134 (2008).

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 \(\wedge ^3 H\), 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 \(\frac{1}{2}\wedge ^3H\rtimes \text{Sp}(H)\). This Johnson-Morita homomorphism is not surjective, but its image is of finite index in \(\frac{1}{2}\wedge ^3H\rtimes \text{Sp}(H)\), cf. S. Morita [Invent. Math. 111, No.1, 197–224 (1993; Zbl 0787.57008 )]. In the present paper the authors give a description of the exact image of Morita’s homomorphism, and they compute the image of the handlebody subgroup of the mapping class group under the same map.
For the entire collection see [Zbl 1154.53004].

Reviewer: Masako Kobayashi (Osaka)

MSC:

57M99	General low-dimensional topology
57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)

Keywords:

mapping class group; Johnson-Morita homomorphism; Torelli subgroup; surface

Citations:

Zbl 0787.57008