Advanced Studies in Pure Mathematics

Calculating the image of the second Johnson-Morita representation

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

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Abstract

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 $\frac{1}{2} \wedge^3 H \rtimes \mathrm{S_p}(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in $\frac{1}{2} \wedge^3 H \rtimes \mathrm{S_p}(H)$ [11]. 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.

Article information

Source
Groups of Diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday, R. Penner, D. Kotschick, T. Tsuboi, N. Kawazumi, T. Kitano and Y. Mitsumatsu, eds. (Tokyo: Mathematical Society of Japan, 2008), 119-134

Dates
Received: 30 April 2007
Revised: 21 August 2007
First available in Project Euclid: 28 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543447482

Digital Object Identifier
doi:10.2969/aspm/05210119

Mathematical Reviews number (MathSciNet)
MR2509709

Zentralblatt MATH identifier
1183.57016

Citation

Birman, Joan S.; Brendle, Tara E.; Broaddus, Nathan. Calculating the image of the second Johnson-Morita representation. Groups of Diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday, 119--134, Mathematical Society of Japan, Tokyo, Japan, 2008. doi:10.2969/aspm/05210119. https://projecteuclid.org/euclid.aspm/1543447482


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