Geometry and Topology Volume 12, issue 2 (2008)

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.

Keywords

3-manifold, finite-type invariant, LMO invariant, Kontsevich integral, cobordism category, Lagrangian cobordism, homology cylinder, bottom-top tangle, Jacobi diagram, clasper

Mathematical Subject Classification

Primary: 57M27

Secondary: 57M25

References

Publication

Received: 28 March 2007
Accepted: 16 January 2008
Published: 24 May 2008
Proposed: Peter Teichner
Seconded: Shigeyuki Morita, Rob Kirby

Authors

Dorin Cheptea
Previous address: Center for the Topology and Quantization of Moduli Spaces University of Aarhus Bygning 1530 Ny Munkegade 8000 Aarhus C Denmark Current address: Institute of Mathematics of the Romanian Academy PO Box 1-764 Bucharest RO - 014700 Romania

Kazuo Habiro
Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan

Gwénaël Massuyeau
Institut de Recherche Mathématique Avancée Université Louis Pasteur – CNRS 7 rue René Descartes 67084 Strasbourg France