Canonical lifts of the Johnson homomorphisms to the Torelli groupoid

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 $m\ge 1$, 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 $m\le 3$.

38 pages, 6 figures

References in corpus (1)

• Harmonic Magnus Expansion on the Universal Family of Riemann Surfaces