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Communications in Mathematical Physics

, Volume 272, Issue 3, pp 601–634 | Cite as

A TQFT Associated to the LMO Invariant of Three-Dimensional Manifolds

  • Dorin ChepteaEmail author
  • Thang T. Q. Le
Article

Abstract

We construct a Topological Quantum Field Theory associated to the universal finite-type invariant of 3-dimensional manifolds, as a functor from a category of 3-dimensional manifolds with parametrized boundary, satisfying some additional conditions, to an algebraic-combinatorial category. This is built together with its truncations with respect to a natural grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The TQFT(s) induce(s) a (series of) representation(s) of a subgroup \({{\mathcal{L}}}_g\) of the Mapping Class Group that contains the Torelli group. The N =  1 truncation is a TQFT for the Casson-Walker-Lescop invariant.

Keywords

Boundary Component Mapping Class Group Chord Diagram Chain Graph Link Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Atiyah M. (1988). Topological Quantum Field Theories. Publications Mathématiques IHES 68: 175–186 zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bar-Natan D. (1995). On the Vassiliev knot invariants. Topology 34: 423–472 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bar-Natan D. and Lawrence R. (2004). A rational surgery formula for the LMO invariant. Israel J. Math. 140: 29–60 zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bar-Natan D., Le T. and Thurston D. (2003). Two applications of elementary knot theory to Lie algebras and Vassiliev invariants. Geom. Topol. 7: 1–31 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Blanchet C., Habegger H., Masbaum G. and Vogel P. (1995). Topological quantum field theories derived from the Kauffman bracket. Topology 34(4): 883–927 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cheptea, D., Le, T.: 3-cobordisms with their rational homology on the boundary. Preprint, available at http://arxiv.org/math/0602097, 2006Google Scholar
  7. 7.
    Cheptea, D., Habiro, K., Massuyeau, G.: A functorial LMO invariant for Lagrangian cobordisms. Preprint, available at http://arxiv.org/math/0701277, 2007Google Scholar
  8. 8.
    Fomenko A. and Matveev S. (1997). Algorithmic and computer methods for three-manifolds. Kluwer Academic Publishers, Dordrecht zbMATHGoogle Scholar
  9. 9.
    Gille C. (2003). On the Le-Murakami-Ohtsuki invariant in degree 2 for several classes of 3-manifolds. J Knot Theory Ramifications 12(1): 17–45 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gompf, R.E., Stipsicz, A.I.: 4-manifolds and Kirby calculus. Graduate Studies in Mathematics 20, Providence, RI: Amer. Math. Soc., 1999Google Scholar
  11. 11.
    Habegger N. and Masbaum G. (2000). The Kontsevich integral and Milnor’s invariants. Topology 39: 1253–1289 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Habegger N. and Orr K. (1999). Finite type three manifold invariants -realization and vanishing. J. Knot Theory Ramifications 8(8): 1001–1007 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Habegger N. and Orr K. (1999). Milnor link invariants and quantum 3-manifold invariants. Comment. Math. Helv. 74(2): 322–344 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Habiro K. (2000). Claspers and finite-type invariants of links. Geom. and Top. 4: 1–83 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Le T.T.Q. (1997). An invariant of integral homology 3-spheres which is universal for all finite type invariants. AMS Translation series 2(179): 75–100 Google Scholar
  16. 16.
    Le, T.T.Q.: The LMO invariant, “Invariants de noeuds at de variétés de dimension 3”. In: Proc. of École d’été de Mathématiques, Grenoble: Institut Fourier, 1999Google Scholar
  17. 17.
    Le T.T.Q., Murakami J. and Ohtsuki T. (1998). On a universal perturbative invariant of 3-manifolds. Topology 37(3): 539–574 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lescop C. (1996). Global surgery formula for the Casson-Walker invariant. Princeton University Press, Princeton, NJ zbMATHGoogle Scholar
  19. 19.
    Matveev S.V. (1986). Generalized surgery of three-dimensional manifolds and representations of homology spheres. Matematicheskie Zametki 42(2): 268–278 Google Scholar
  20. 20.
    Morita S. (1989). Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I. Topology 28(3): 305–323 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Murakami J. and Ohtsuki T. (1997). Topological Quantum Field Theory for the Universal Quantum Invariant. Commun. Math. Phys. 188: 501–520 zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Serre, J-P.: Lie Algebras and Lie Groups, 2nd ed., Lecture Notes in Mathematics 1500, New York: Springer, 1992Google Scholar
  23. 23.
    Turaev V. (1994). Quantum Invariants of Knots and 3-Manifolds. Walter de Gruyter, Berlin zbMATHGoogle Scholar
  24. 24.
    Vogel, P.: Invariants de type fini. In: “Nouveaux Invariants en Géométrie et en Topologie”, publié par D. Bennequin, M. Audin, J. Morgan, P. Vogel, Panoramas et Synthèses 11, Paris: Société Mathématique de France, 2001 pp. 99–128Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Center for the Topology and Quantization of Moduli Spaces, Department of Mathematical SciencesUniversity of AarhusAarhus CDenmark
  2. 2.Institute of Mathematics of the Romanian AcademyBucharestRomania
  3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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