Canonical lifts of the Johnson homomorphisms to the Torelli groupoid

Bene, Alex James; Kawazumi, Nariya; Penner, R. C.

Canonical lifts of the Johnson homomorphisms to the Torelli groupoid

Authors

Bene, Alex James

Kawazumi, Nariya

Penner, R. C.

Description

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 $τ_m$, 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 $τ_2$ 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 $τ_m$, for $m\leq 3$.
38 pages, 6 figures