Canonical lifts of the Johnson homomorphisms to the Torelli groupoid

TL;DR

This paper constructs canonical lifts of Johnson homomorphisms to the Torelli groupoid using Magnus expansions derived from fatgraph structures, providing explicit formulas and examples for these lifts.

Contribution

It introduces a canonical Magnus expansion for fatgraphs and uses it to explicitly lift Johnson homomorphisms to the Torelli groupoid, answering a longstanding question.

Findings

Constructed a canonical Magnus expansion for fatgraphs.

Provided explicit recursive formulas for Johnson homomorphism lifts.

Calculated examples of lifts for m ≤ 3.

Abstract

We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical lifts of the higher Johnson homomorphisms $\tau_m$, for $m\geq 1$, to the Torelli groupoid, and we provide a recursive combinatorial formula for tensor representatives of these lifts. In particular, we give an explicit 1-cocycle in the dual fatgraph complex which lifts $\tau_2$ and thus answer affirmatively a question of Morita-Penner. To illustrate our techniques for calculating higher Johnson homomorphisms in general, we give explicit examples calculating $\tau_m$, for $m\leq 3$.