Canonical lifts of the Johnson homomorphisms to the Torelli groupoid

Details Canonical lifts of the Johnson homomorphisms to the Torelli groupoid Canonical lifts of the Johnson homomorphisms to the Torelli groupoid

2007-07-20

arxiv.org/abs/0707.2984v1

Mathematics

Geometric Topology

38 pages, 6 figures

Scientific paper

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 $m\leq 3$.

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