Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces

Shigeyuki Morita

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
Revised: 9 January 2008
First available in Project Euclid: 28 November 2018

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