A functorial LMO invariant for Lagrangian cobordisms

Cheptea, Dorin¹ ; Habiro, Kazuo² ; Massuyeau, Gwénaël³

¹ 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
² Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
³ Institut de Recherche Mathématique Avancée, Université Louis Pasteur – CNRS, 7 rue René Descartes, 67084 Strasbourg, France

Geometry & topology, Tome 12 (2008) no. 2, pp. 1091-1170.

Résumé

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.

Détail

DOI : 10.2140/gt.2008.12.1091

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

Bibliographie

[1] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423

[2] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The \AArhus integral of rational homology 3-spheres. I. A highly non trivial flat connection on $S^3$, Selecta Math. $($N.S.$)$ 8 (2002) 315

[3] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, The \AArhus integral of rational homology 3-spheres. II. Invariance and universality, Selecta Math. $($N.S.$)$ 8 (2002) 341

[4] D Bar-Natan, R Lawrence, A rational surgery formula for the LMO invariant, Israel J. Math. 140 (2004) 29

[5] D Cheptea, T T Q Le, Three-cobordisms with their rational homology on the boundary, preprint (2005) to appear in J. Knot Theory Ramifications

[6] D Cheptea, T T Q Le, A TQFT associated to the LMO invariant of three-dimensional manifolds, Comm. Math. Phys. 272 (2007) 601

[7] L Crane, D Yetter, On algebraic structures implicit in topological quantum field theories, J. Knot Theory Ramifications 8 (1999) 125

[8] S Garoufalidis, The mystery of the brane relation, J. Knot Theory Ramifications 11 (2002) 725

[9] S Garoufalidis, M Goussarov, M Polyak, Calculus of clovers and finite type invariants of 3–manifolds, Geom. Topol. 5 (2001) 75

[10] S Garoufalidis, J Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, from: "Graphs and patterns in mathematics and theoretical physics", Proc. Sympos. Pure Math. 73, Amer. Math. Soc. (2005) 173

[11] M Goussarov, Finite type invariants and $n$–equivalence of $3$–manifolds, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 517

[12] M Goussarov, Variations of knotted graphs. The geometric technique of $n$–equivalence, Algebra i Analiz 12 (2000) 79

[13] N Habegger, Milnor, Johnson and tree-level perturbative invariants, preprint (2000)

[14] N Habegger, X S Lin, On link concordance and Milnor's $\overline {\mu}$ invariants, Bull. London Math. Soc. 30 (1998) 419

[15] N Habegger, G Masbaum, The Kontsevich integral and Milnor's invariants, Topology 39 (2000) 1253

[16] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1

[17] K Habiro, Bottom tangles and universal invariants, Algebr. Geom. Topol. 6 (2006) 1113

[18] D M Jackson, I Moffatt, A Morales, On the group-like behavior of the Le–Murakami–Ohtsuki invariant, J. Knot Theory Ramifications 16 (2007) 699

[19] T Kerler, Bridged links and tangle presentations of cobordism categories, Adv. Math. 141 (1999) 207

[20] T Kerler, Towards an algebraic characterization of 3–dimensional cobordisms, from: "Diagrammatic morphisms and applications (San Francisco, CA, 2000)", Contemp. Math. 318, Amer. Math. Soc. (2003) 141

[21] T T Q Le, An invariant of integral homology $3$–spheres which is universal for all finite type invariants, from: "Solitons, geometry, and topology: on the crossroad", Amer. Math. Soc. Transl. Ser. 2 179, Amer. Math. Soc. (1997) 75

[22] T Q T Le, The LMO invariant, lecture notes from: “Ecole d’Eté de Mathématiques: Invariants de noeuds et de variétés de dimension trois”, Institut Fourier (1999)

[23] T Q T Le, J Murakami, Representation of the category of tangles by Kontsevich's iterated integral, Comm. Math. Phys. 168 (1995) 535

[24] T Q T Le, J Murakami, The universal Vassiliev–Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996) 41

[25] T T Q Le, J Murakami, Parallel version of the universal Vassiliev–Kontsevich invariant, J. Pure Appl. Algebra 121 (1997) 271

[26] T T Q Le, J Murakami, T Ohtsuki, On a universal perturbative invariant of $3$–manifolds, Topology 37 (1998) 539

[27] C Lescop, Splitting formulae for the Kontsevich–Kuperberg–Thurston invariant of rational homology $3$–spheres, preprint (2004)

[28] G Massuyeau, Finite-type invariants of 3–manifolds and the dimension subgroup problem, J. Lond. Math. Soc. $(2)$ 75 (2007) 791

[29] S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268, 345

[30] J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. of Math. $(2)$ 81 (1965) 211

[31] I Moffatt, The \AArhus integral and the $\mu$–invariants, J. Knot Theory Ramifications 15 (2006) 361

[32] S Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991) 603

[33] H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75

[34] J Murakami, T Ohtsuki, Topological quantum field theory for the universal quantum invariant, Comm. Math. Phys. 188 (1997) 501

[35] T Ohtsuki, Quantum invariants, Series on Knots and Everything 29, World Scientific Publishing Co. (2002)

[36] J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170