Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant

By G. Massuyeau. 2008

66 citations recorded by the source corpus.

Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quotient of P. Morita defined a homomorphism from the k-th term of the Johnson filtration to the third homology group of the k-th nilpotent quotient of P. In this paper, we replace groups by their Malcev Lie algebras and we study the "infinitesimal" version of the k-th Morita homomorphism, which is shown to correspond to the original version by a canonical isomorphism. We provide a diagrammatic description of the k-th infinitesimal Morita homomorphism and, given an expansion of the free group P that is "symplectic" in some sense, we show how to compute it from Kawazumi's "total Johnson map". Besides, we give a topological interpretation of the full tree-reduction of the LMO homomorphism, which is a diagrammatic representation of the Torelli group derived from the Le-Murakami-Ohtsuki invariant of 3-manifolds. More precisely, a symplectic expansion of P is constructed from the LMO invariant, and it is shown that the tree-level of the LMO homomorphism is equivalent to the total Johnson map induced by this specific expansion. It follows that the k-th infinitesimal Morita homomorphism coincides with the degree [k,2k[ part of the tree-reduction of the LMO homomorphism. Our results also apply to the monoid of homology cylinders over S.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a4b92d3288109cde9dc39c5d5be779263bb9076a

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In Geometry and Topology

1. Y. Kuno
2. Nariya Kawazumi
3. G. Massuyeau
4. Alexandra Pettet

In Mathematical Physics

1. Y. Kuno
2. Nariya Kawazumi
3. G. Massuyeau
4. Alexandra Pettet

Citations in the corpus

• The logarithms of Dehn twists (2010, methodology): “It naturally has a structure of a complete augmented algebra (in the sense of Quillen [25], Appendix A) with respect to a decreasing filtration given by lim←−m≥p Iπ p/Iπm, for p ≥ 1.”