## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- 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

### Calculating the image of the second Johnson-Morita representation

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

#### 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**

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