## 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), 443 - 468

### Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces

#### Abstract

We describe the *group version* of the trace maps given in [29]. This gives rise to abelian quotients of *symplectic* IA-automorphism groups of nilpotent quotients of the fundamental groups of compact surfaces. By making use of them, we construct a representation of the group $\mathcal{H}_{g,1}$ of homology cobordism classes of homology cylinders introduced by Garoufalidis and Levine [6]. We define various cohomology classes of $\mathcal{H}_{g,1}$ and propose a few problems concerning them. In particular, we mention a possible relation to additive invariants for the group $\Theta_{\mathbb{Z}}^{3}$ of homology cobordism classes of homology 3-spheres.

#### Article information

**Dates**

Received: 15 October 2007

Revised: 9 January 2008

First available in Project Euclid:
28 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543447493

**Digital Object Identifier**

doi:10.2969/aspm/05210443

**Mathematical Reviews number (MathSciNet)**

MR2509720

**Zentralblatt MATH identifier**

1166.57012

#### Citation

Morita, Shigeyuki. Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces. Groups of Diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday, 443--468, Mathematical Society of Japan, Tokyo, Japan, 2008. doi:10.2969/aspm/05210443. https://projecteuclid.org/euclid.aspm/1543447493