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Invariants of piecewise-linear 3-manifolds

Authors: John W. Barrett and Bruce W. Westbury
Journal: Trans. Amer. Math. Soc. 348 (1996), 3997-4022
MSC (1991): Primary 57N10
MathSciNet review: 1357878
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Abstract: This paper presents an algebraic framework for constructing invariants of closed oriented 3-manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A significant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity.

References [Enhancements On Off] (What's this?)

  • [1] H. H. Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992), 149--159. MR 94b:17015
  • [2] H. H. Andersen and J. Paradowski, Fusion categories arising from semisimple Lie algebras, Comm. Math. Phys. 169 (1995), 563--588. MR 96e:17026
  • [3] J. W. Barrett and B. W. Westbury, Spherical categories, preprint, hep-th/9310164, University of Nottingham, 1993.
  • [4] J. W. Barrett and B. W. Westbury, The equality of 3-manifold invariants, Math. Proc. Cambridge Philos. Soc. 118 (1995), 503--510.
  • [5] B. Durhuus, H. P. Jakobsen and R. Nest, Topological quantum field theories from generalized $6j$-symbols, Reviews in Math. Physics 5 (1993), 1--67. MR 94h:57025
  • [6] P. J. Freyd and D. N. Yetter, Braided compact closed categories with applications to low dimensional topology, Adv. in Math. 77 (1989), 156--182. MR 91c:57019
  • [7] P. Freyd and D. N. Yetter, Coherence theorems via knot theory, J. Pure Appl. Algebra 78 (1992), 49--76. MR 93d:18013
  • [8] L. C. Glaser, Geometrical Combinatorial Topology I, Van Nostrand Reinhold Mathematical Studies 27 (1970).
  • [9] A. Joyal and R. Street, The geometry of tensor calculus, I, Adv. in Math. 88 (1991), 55--112. MR 92d:18011
  • [10] G. M. Kelly and M. I. Laplaza, Coherence for compact closed categories, J. Pure Appl. Algebra 19 (1980), 193--213. MR 81m:18008
  • [11] G. Kuperberg, Involutory Hopf algebras and 3-manifold invariants, International Journal of Mathematics 2 (1991), 41--66. MR 91m:57012
  • [12] R. G. Larson and D. E. Radford, Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra (1988), 267--289. MR 89k:16016
  • [13] R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 75--94. MR 39:1523
  • [14] J. P. Moussouris, Quantum models of space-time based on coupling theory, D. Phil. Oxford (1983).
  • [15] U. Pachner, P.l. homeomorphic manifolds are equivalent by elementary shellings, European Journal of Combinatorics 12 (1991), 129--145. MR 92d:52040
  • [16] G. Ponzano and T. Regge, Semiclassical limit of Racah coefficients, in Spectroscopic and Group Theoretical Methods in Physics, North-Holland, Amsterdam, 1968, pp. 1--58.
  • [17] N. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547--597. MR 92b:57024
  • [18] N. Y. Reshetikhin and A. N. Kirillov, Representations of the algebra $U_q(\mathrm {sl}(2))$, $q$-orthogonal polynomials and invariants of links in Infinite-dimensional Lie Algebras and Groups, (V. G. Kac, ed.), World Scientific, Singapore, 1988, pp. 285--339. MR 90m:17022
  • [19] N. Y. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1--26. MR 91c:57016
  • [20] J. Roberts, Skein theory and Turaev-Viro invariants, Topology 34 (1995), 771--787.
  • [21] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, New York--Heidelberg--Berlin, 1982. MR 83g:57009
  • [22] V. Turaev, Quantum invariants of 3-manifold and a glimpse of shadow topology, in Quantum Groups (Leningrad, 1980), Lecture Notes in Math. 1510, Springer-Verlag, Berlin, 1992, pp. 363--366. MR 93j:57010
  • [23] V. G. Turaev, Modular categories and 3-manifold invariants, Internat. J. Modern Phys. 6 (1992), 1807--1824. MR 93k:57040
  • [24] V. G. Turaev, Quantum Invariants of Knots and 3-manifolds, De Gruyter, New York--Berlin, 1994. MR 95k:57014
  • [25] V. G. Turaev and O. Y. Viro, State sum invariants of 3-manifolds and quantum $6j$-symbols, Topology 31 (1992), 865--902. MR 94d:57044
  • [26] V. Turaev and H. Wenzl, Quantum invariants of 3-manifolds associated with classical simple Lie algebras, International Journal of Mathematics 4 (1993), 323--358. MR 94i:57019
  • [27] K. Walker, On Witten's 3-manifold invariants, 1990.
  • [28] D. N. Yetter, State-sum invariants of 3-manifolds associated to Artinian semisimple tortile categories, Topology and its Applications 58 (1993), 47--80. MR 95g:57032

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Additional Information

John W. Barrett
Affiliation: Department of Mathematics, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K.

Bruce W. Westbury
Affiliation: Department of Mathematics, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K.

Keywords: 3-manifold invariants, state sum model, bistellar moves, spherical category, spherical Hopf algebra
Received by editor(s): July 20, 1994
Article copyright: © Copyright 1996 American Mathematical Society

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