Strong integrality of quantum invariants of 3manifolds
Author:
Thang T. Q. Lê
Journal:
Trans. Amer. Math. Soc. 360 (2008), 29412963
MSC (2000):
Primary 57M27; Secondary 57M25
Published electronically:
December 11, 2007
MathSciNet review:
2379782
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Abstract: We prove that the quantum invariant of an arbitrary 3manifold is always an algebraic integer if the order of the quantum parameter is coprime with the order of the torsion part of . An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3spheres. Here we also generalize Habiro's result to all rational homology 3spheres.
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 [BBL]
 A. Beliakova, C. Blanchet, and T. T. Q. Le, Laplace transform and universal sl(2) invariants, preprint math.QA/0509394.
 [GM]
 P. Gilmer and G. Masbaum, Integral lattices in TQFTs, preprint math.QA/0411029.
 [Ha1]
 K. Habiro, On the quantum invariants of knots and integral homology spheres, Geom. Topol. Monogr. 4 (2002) 5568. MR 2002603 (2004g:57023)
 [Ha2]
 K. Habiro, Cyclotomic completions of polynomial rings, Publ. Res. Inst. Math. Sci. 40 (2004), 11271146. MR 2105705 (2005j:13010)
 [Ha3]
 K. Habiro, An integral form of the quantized enveloping algebra of and its completions, J. Pure Appl. Algebra 211 (2007), 265292. MR 2333771
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 K. Habiro and T. T. Q. Lê, in preparation.
 [KM]
 R. Kirby and P. Melvin, The manifold invariants of Witten and ReshetikhinTuraev for , Invent. Math. 105 (1991), 473545. MR 1117149 (92e:57011)
 [KK]
 A. Kawauchi and S. Kojima, Algebraic classification of linking pairings on manifolds, Math. Ann. 253 (1980), 2942. MR 594531 (82b:57007)
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 V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335388. MR 908150 (89c:46092)
 [Le1]
 T. T. Q. Lê, An invariant of integral homology 3spheres which is universal for all finite type invariants, AMS translation series 2, Eds. V. Buchtaber and S. Novikov, 179 (1997), 75100. MR 1437158 (99m:57008)
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 [Le3]
 T. T. Q. Lê, Quantum invariants of 3manifolds: integrality, splitting, and perturbative expansion, Topology Appl. 127 (2003), 125152. MR 1953323 (2005b:57026)
 [LL]
 B.H. Li and T.J. Li, Generalized Gaussian sums: ChernSimonsWittenJones invariants of lensspaces, J. Knot Theory Ramifications 5 (1996), 183224. MR 1395779 (97e:57018)
 [Ma]
 G. Masbaum, Skeintheoretical derivation of some formulas of Habiro, Algebraic and Geometric Topology 3 (2003), 537556. MR 1997328 (2004f:57013)
 [MR]
 G. Masbaum and J. Roberts, A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Camb. Phil. Soc. 121 (1997), 443454. MR 1434653 (98h:57037)
 [MW]
 G. Masbaum and H. Wenzl, Integral modular categories and integrality of quantum invariants at roots of unity of prime order, J. Reine Angew. Math. 505 (1998), 209235. MR 1662260 (2000a:57043)
 [Mu]
 H. Murakami, Quantum invariants dominate the invariant of Casson and Walker, Math. Proc. Camb. Phil. Soc. 117 (1995), 237249. MR 1307078 (95k:57027)
 [LMO]
 T. T. Q. Lê, J. Murakami, and T. Ohtsuki, On a universal perturbative invariant of manifolds, Topology 37 (1998), 539574. MR 1604883 (99d:57004)
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 T. Ohtsuki, A polynomial invariant of rational homology spheres, Invent. Math. 123 (1996), 241257. MR 1374199 (97i:57018)
 [Oht2]
 T. Ohtsuki, Quantum invariants. A study of knots, 3manifolds, and their sets, Series on Knots and Everything, 29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. MR 1881401 (2003f:57027)
 [RG]
 H. Rademacher and E. Grosswald, Dedekind Sums, Amer. Math. Soc., Washington D. C., 1972. MR 0357299 (50:9767)
 [Ros]
 L. Rozansky, On padic properties of the WittenReshetikhinTuraev invariant, preprint math.QA/9806075, 1998.
 [TY]
 T. Takata and Y. Yokota, The invariants of 3manifolds are polynomials, J. Knot Theory Ramifications 8 (1999), no. 4, 521532. MR 1697388 (2000k:57016)
 [Tur]
 V. G. Turaev, Quantum invariants of knots and 3manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter, Berlin, New York, 1994. MR 1292673 (95k:57014)
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 C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 1963, 281298 MR 0156890 (28:133)
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Additional Information
Thang T. Q. Lê
Affiliation:
Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 303320160
Email:
letu@math.gatech.edu
DOI:
http://dx.doi.org/10.1090/S0002994707043590
PII:
S 00029947(07)043590
Received by editor(s):
March 2, 2006
Published electronically:
December 11, 2007
Additional Notes:
The author was supported in part by the National Science Foundation
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
