Homology TQFT's and the Alexander-Reidemeister invariant of 3-manifolds via Hopf algebras and Skein theory

Canadian Journal of Mathematics

Volumn 55, Issue 4, 2003, Pages 766-821

  Kerler, Thomas ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0043076386     PISSN: 0008414X     EISSN: None     Source Type: Journal    
DOI: 10.4153/CJM-2003-033-5     Document Type: Article

References (53)

1. M. Atiyah, Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. 68 (1988), 175-186.
2. H. Boden and A. Nicas, Universal formulae for SU(n) Casson invariants of knots. Trans. Amer. Math. Soc. (7) 352(2000), 3149-3187.
3. Yu. Bespalov, T. Kerler, V. Lyubashenko and V. Turaev, Integrals for braided Hopf algebras. J. Pure Appl. Algebra (2) 148(2000), 113-164.
4. G. Burde and H. Zieschang, Knots. de Gruyter Stud. Math. 5, Walter de Gruyter, 1985.
5. S. K. Donaldson, Topological field theories and formulae of Casson and Meng - Taubes. In: Proceedings of the Kirbyfest (Berkeley, CA), Geom. Topol. Monogr. 2(1999), 87-102 (electronic).
6. V. G. Drinfeld, On almost cocommutative Hopf algebras. Leningrad Math. J. (2) 1(1990), 321- 342.
7. C. Frohman, Unitary representations of knot groups. Topology (1) 32(1993), 121-144.
8. C. Frohman and A. Nicas, The Alexander Polynomial via topological quantum field theory. In: Differential Geometry, Global Analysis, and Topology, CMS Conf. Proc. 12, Amer. Math. Soc. Providence, RI, 1991, 27-40.
9. _, An intersection homology invariant for knots in a rational homology 3-sphere. Topology (1) 33(1994), 123-158.
10. V. V. Fock and A. A. Rosly, Moduli space of flat connections as a Poisson manifold. In: Advances in quantum field theory and statistical mechanics, 2nd Italian - Russian collaboration (Como, 1996), Internat. J. Modern Phys. B 11(1997), 3195-3206.
11. P. Gilmer, Integrality for TQFTs. Preprint, 2001, math.QA/0105059.
12. R. Goodman and N. R. Wallach, Representations and invariants of the classical groups. Encyclopedia Math. Appl. 68, Cambridge University Press, 1998.
13. P. Griffith and J. Harris, Principles of algebraic geometry. Wiley Classics Library, John Wiley & Sons, Inc., New York, 1978 and 1994.
14. M. Hennings, Invariants of links and 3-manifolds obtained from Hopf algebras. J. London Math. Soc. (2) 54(1996), 594-624.
15. L. H. Kauffman and D. E. Radford, Invariants of 3-manifolds derived from finite-dimensional Hopf algebras. J. Knot Theory Ramifications (1) 4(1995), 131-162.
16. L. H. Kauffman, Gauss codes, quantum groups and ribbon Hopf algebras. Rev. Math. Phys. (4) 5(1993), 735-773.
17. T. Kerler, Darstellungen der Quantengruppen und Anwendungen. Diploma thesis, ETH- Zürich, 1989, unpublished.
18. _, Mapping class group actions on quantum doubles. Comm. Math. Phys. 168(1994), 353- 388.
19. _, Genealogy of nonperturbative quantum-invariants of 3-manifolds: The surgical family. In: Geometry and Physics, Lecture Notes in Pure and Appl. Physics 184, Marcel Dekker, 1997, 503-547.
20. _, On the connectivity of cobordisms and half-projective TQFT's. Comm. Math. Phys. (3) 198(1998), 535-590.
21. _, Bridged links and tangle presentations of cobordism categories. Adv. Math. 141(1999), 207-281.
22. _, Resolutions of p-Modular TQFT's and Representations of Symmetric Groups. math.GT/0110006.
23. _, The Structure of the Fibonacci TQFT. In preparation.
24. _, p-Modular TQFT's, Milnor-Torsion, and the Casson-Lescop Invariant. Geom. Topol. Monogr. 4(2002), 119-141.
25. T. Kerler and V. V. Lyubashenko, Non-semisimple topological quantum field theories for 3- manifolds with corners. Lecture Notes in Math., Springer-Verlag, 2001, to appear.
26. 3. Invent. Math. 45(1978), 35-56.
27. G. Kuperberg, Noninvolutory Hopf algebras and 3-manifold invariants. Duke Math. J. (1) 84 (1996), 83-129; Involutory Hopf algebras and 3-manifold invariants. Internat. J. Math. (1) 2(1991), 41-66.
28. G. Kuperberg, Noninvolutory Hopf algebras and 3-manifold invariants. Duke Math. J. (1) 84 (1996), 83-129; Involutory Hopf algebras and 3-manifold invariants. Internat. J. Math. (1) 2(1991), 41-66.
29. R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91(1969), 75-94.
30. C. Lescop, Global surgery formula for the Casson-Walker invariant. Ann. of Math. Stud. 140, Princeton University Press, Princeton, NJ, 1996.
31. W. B. R. Lickorish, An Introduction to Knot Theory. Graduate Texts in Math. 127, Springer Verlag, 1997.
32. V. V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity. Comm. Math. Phys. 172(1995), 467-516.
33. V. V. Lyubashenko and S. Majid, Braided groups and quantum Fourier transform. J. Algebra (3) 166(1994), 506-528.
34. S. Majid, Braided groups. J. Pure Appl. Algebra (2) 86(1993), 187-221.
35. W. S. Massey, A Basic Course in Algebraic Topology. Graduate Texts in Math. 127, Springer Verlag, 1991.
36. G. Masbaum and J. D. Roberts, A simple proof of integrality of quantum invariants at prime roots of unity. Math. Proc. Cambridge Philos. Soc. (3) 121(1997), 443-454.
37. S. Matveev and M. Polyak, A geometrical presentation of the surface mapping class group and surgery. Comm. Math. Phys. 160(1994), 537-550.
38. G. Meng and C. Taubes, SW = Milnor torsion. Math. Res. Lett. (5) 3(1996), 661-674.
39. J. Milnor, A duality theorem for Reidemeister torsion. Ann. of Math. (2) 76(1962), 137-147.
40. H. Murakami, Quantum SU(2)-invariants dominate Casson's SU(2)-invariant. Math. Proc. Camb. Phil. Soc. 115(1994), 83-103.
41. T. Ohtsuki, A polynomial invariant of rational homology 3-spheres. Invent. Math. (2) 123 (1996), 241-257.
42. D. Radford, The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite. Amer. J. Math. (2) (1976)98, 333-355.
43. _, The trace function and Hopf algebras. J. Algebra 163(1994), 583-622.
44. N. Yu. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(1991), 547-597.
45. L. Rozansky and H. Saleur, S- and T -matrices for the super U(1,1)WZW model. Application to surgery and 3-manifolds invariants based on the Alexander-Conway polynomial. Nuclear Phys. B (2) 389(1993), 365-423.
46. V. G. Turaev, Reidemeister torsion and the Alexander polynomial. (Russian) Mat. Sb. (N.S.) (2) 18(66)(1976), 252-270.
47. _, Quantum invariants of knots and 3-manifolds. de Gruyter Stud. Math. 18, Walter de Gruyter, Berlin, 1994.
48. _, A combinatorial formulation for the Seiberg-Witten invariants of 3-manifolds. Math. Res. Lett. (5) 5(1998), 583-598.
49. V. Turaev and O. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols. Topology (4) 31(1992), 865-902.
50. B. Wajnryb, An elementary approach to the mapping class group of a surface. Geom. Topol. 3 (1999), 405-466 (electronic); A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45(1983), 157-174.
51. B. Wajnryb, An elementary approach to the mapping class group of a surface. Geom. Topol. 3 (1999), 405-466 (electronic); A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45(1983), 157-174.
52. E. Witten, Quantum field theory and the Jones polynomial Comm. Math. Phys. (3) 121(1989), 351-399.
53. D. Yetter, Portrait of the handle as a Hopf algebra. In: Geometry and physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math. 184, Dekker, New York, 1997, 481-502.