Calculating the image of the second Johnson-Morita representation

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

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

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