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

Gwénaël Massuyeau

Bulletin de la Société Mathématique de France (2012)

  • Volume: 140, Issue: 1, page 101-161
  • ISSN: 0037-9484

Abstract

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Let Σ be a compact connected oriented surface with one boundary component, and let π be the fundamental group of Σ . The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of Σ , whose k -th term consists of the self-homeomorphisms of Σ that act trivially at the level of the k -th nilpotent quotient of π . 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 π . 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 π 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 π 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 , 2 k [ part of the tree-reduction of the LMO homomorphism. Our results also apply to the monoid of homology cylinders over Σ .

How to cite

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Massuyeau, Gwénaël. "Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant." Bulletin de la Société Mathématique de France 140.1 (2012): 101-161. <http://eudml.org/doc/272698>.

@article{Massuyeau2012,
abstract = {Let $\Sigma $ be a compact connected oriented surface with one boundary component, and let $\pi $ be the fundamental group of $\Sigma $. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of $\Sigma $, whose $k$-th term consists of the self-homeomorphisms of $\Sigma $ that act trivially at the level of the $k$-th nilpotent quotient of $\pi $. 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 $\pi $. 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 $\pi $ 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 $\pi $ 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 $\Sigma $.},
author = {Massuyeau, Gwénaël},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Torelli group; Johnson homomorphisms; Morita homomorphisms; Magnus expansion; Malcev Lie algebra; homology cylinder; finite-type invariant; LMO invariant},
language = {eng},
number = {1},
pages = {101-161},
publisher = {Société mathématique de France},
title = {Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant},
url = {http://eudml.org/doc/272698},
volume = {140},
year = {2012},
}

TY - JOUR
AU - Massuyeau, Gwénaël
TI - Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant
JO - Bulletin de la Société Mathématique de France
PY - 2012
PB - Société mathématique de France
VL - 140
IS - 1
SP - 101
EP - 161
AB - Let $\Sigma $ be a compact connected oriented surface with one boundary component, and let $\pi $ be the fundamental group of $\Sigma $. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of $\Sigma $, whose $k$-th term consists of the self-homeomorphisms of $\Sigma $ that act trivially at the level of the $k$-th nilpotent quotient of $\pi $. 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 $\pi $. 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 $\pi $ 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 $\pi $ 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 $\Sigma $.
LA - eng
KW - Torelli group; Johnson homomorphisms; Morita homomorphisms; Magnus expansion; Malcev Lie algebra; homology cylinder; finite-type invariant; LMO invariant
UR - http://eudml.org/doc/272698
ER -

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