Canonical extensions of Morita homomorphisms to the Ptolemy groupoid

TL;DR

This paper extends Johnson's and Morita's homomorphisms to the Ptolemy groupoid of a surface, providing canonical, finitely generated abelian group-valued extensions based on 3D interpretations and homological chain maps.

Contribution

It introduces canonical extensions of homomorphisms to the Ptolemy groupoid, utilizing 3D topology and homological algebra techniques.

Findings

Extensions are canonical and finitely generated.

Values lie in finitely generated free abelian groups.

Based on 3D interpretation and chain maps.

Abstract

Let S be a compact connected oriented surface with one boundary component. We extend each of Johnson's and Morita's homomorphisms to the Ptolemy groupoid of S. Our extensions are canonical and take values into finitely generated free abelian groups. The constructions are based on the 3-dimensional interpretation of the Ptolemy groupoid, and a chain map introduced by Suslin and Wodzicki to relate the homology of a nilpotent group to the homology of its Malcev Lie algebra.