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A functorial LMO invariant for Lagrangian cobordisms

Dorin Cheptea, Kazuo Habiro and Gwénaël Massuyeau

Geometry & Topology 12 (2008) 1091–1170

Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le–Murakami–Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prove some properties of this functorial LMO invariant, including its universality among rational finite-type invariants of Lagrangian cobordisms. Finally, we apply the LMO functor to the study of homology cylinders from the point of view of their finite-type invariants.

3-manifold, finite-type invariant, LMO invariant, Kontsevich integral, cobordism category, Lagrangian cobordism, homology cylinder, bottom-top tangle, Jacobi diagram, clasper
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25
Received: 28 March 2007
Accepted: 16 January 2008
Published: 24 May 2008
Proposed: Peter Teichner
Seconded: Shigeyuki Morita, Rob Kirby
Dorin Cheptea
Previous address: Center for the Topology and Quantization of Moduli Spaces
University of Aarhus
Bygning 1530
Ny Munkegade
8000 Aarhus C
Current address: Institute of Mathematics of the Romanian Academy
PO Box 1-764
RO - 014700
Kazuo Habiro
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502
Gwénaël Massuyeau
Institut de Recherche Mathématique Avancée
Université Louis Pasteur – CNRS
7 rue René Descartes
67084 Strasbourg