Congruence and quantum invariants of 3–manifolds

Gilmer, Patrick M¹

¹Department of Mathematics, Louisiana State University, Baton Rouge LA 70803, USA

Algebraic and Geometric Topology, Tome 7 (2007) no. 4, pp. 1767-1790

Résumé

Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3–manifolds generated by Dehn surgery with denominator f: weak f–congruence, f–congruence, and strong f–congruence. If f is odd, weak f–congruence preserves the ring structure on cohomology with ℤf–coefficients. We show that strong f–congruence coincides with a relation previously studied by Lackenby. Lackenby showed that the quantum SU(2) invariants are well-behaved under this congruence. We strengthen this result and extend it to the SO(3) quantum invariants. We also obtain some corresponding results for the coarser equivalence relations, and for quantum invariants associated to more general modular categories. We compare S3, the Poincaré homology sphere, the Brieskorn homology sphere Σ(2,3,7) and their mirror images up to strong f–congruence. We distinguish the weak f–congruence classes of some manifolds with the same ℤf–cohomology ring structure.

Détail

DOI : 10.2140/agt.2007.7.1767

Keywords: surgery, framed link, modular category, TQFT

Bibliographie

[1] N A’Campo, TQFT, computations and experiments,

[2] B Bakalov, A Kirillov Jr., Lectures on tensor categories and modular functors, University Lecture Series 21, American Mathematical Society (2001)

[3] P Bantay, The kernel of the modular representation and the Galois action in RCFT, Comm. Math. Phys. 233 (2003) 423

[4] C Blanchet, N Habegger, G Masbaum, P Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992) 685

[5] C Blanchet, N Habegger, G Masbaum, P Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995) 883

[6] Q Chen, T Le, Quantum invariants of periodic links and periodic 3–manifolds, Fund. Math. 184 (2004) 55

[7] T D Cochran, A Gerges, K Orr, Dehn surgery equivalence relations on 3–manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 97

[8] T D Cochran, P Melvin, Quantum cyclotomic orders of 3–manifolds, Topology 40 (2001) 95

[9] M K Dabkowski, J H Przytycki, Unexpected connections between Burnside groups and knot theory, Proc. Natl. Acad. Sci. USA 101 (2004) 17357

[10] R H Fox, Congruence classes of knots, Osaka Math. J. 10 (1958) 37

[11] M Freedman, V Krushkal, On the asymptotics of quantum $\mathrm{SU}(2)$ representations of mapping class groups, Forum Math. 18 (2006) 293

[12] T Gannon, Moonshine beyond the Monster, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2006)

[13] P M Gilmer, On the Witten–Reshetikhin–Turaev representations of mapping class groups, Proc. Amer. Math. Soc. 127 (1999) 2483

[14] P M Gilmer, Integrality for TQFTs, Duke Math. J. 125 (2004) 389

[15] P Gilmer, G Masbaum, Integral lattices in TQFT, Ann. Sci. École Norm. Sup. (to appear)

[16] P M Gilmer, G Masbaum, P Van Wamelen, Integral bases for TQFT modules and unimodular representations of mapping class groups, Comment. Math. Helv. 79 (2004) 260

[17] R E Gompf, A I Stipsicz, 4–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999)

[18] N Jacobson, Basic algebra II, W H Freeman and Company (1989)

[19] L H Kauffman, S L Lins, Temperley–Lieb recoupling theory and invariants of 3–manifolds, Annals of Mathematics Studies 134, Princeton University Press (1994)

[20] M Lackenby, Fox's congruence classes and the quantum–$\mathrm{SU}(2)$ invariants of links in 3–manifolds, Comment. Math. Helv. 71 (1996) 664

[21] S Lang, $\mathrm{SL}_{2}(\mathbb{R})$, Addison–Wesley (1975)

[22] M Larsen, Z Wang, Density of the SO(3) TQFT representation of mapping class groups, Comm. Math. Phys. 260 (2005) 641

[23] W B R Lickorish, The unknotting number of a classical knot, from: "Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982)", Contemp. Math. 44, Amer. Math. Soc. (1985) 117

[24] G Masbaum, J D Roberts, On central extensions of mapping class groups, Math. Ann. 302 (1995) 131

[25] G Masbaum, J D Roberts, A simple proof of integrality of quantum invariants at prime roots of unity, Math. Proc. Cambridge Philos. Soc. 121 (1997) 443

[26] J Milnor, On the 3–dimensional Brieskorn manifolds $M(p,q,r)$, from: "Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox)", Ann. of Math. Studies 84, Princeton Univ. Press (1975) 175

[27] J M Montesinos, Seifert manifolds that are ramified two-sheeted cyclic coverings, Bol. Soc. Mat. Mexicana $(2)$ 18 (1973) 1

[28] H Murakami, Quantum $\mathrm{SO}(3)$–invariants dominate the $\mathrm{SU}(2)$–invariant of Casson and Walker, Math. Proc. Cambridge Philos. Soc. 117 (1995) 237

[29] M Neuhauser, An explicit construction of the metaplectic representation over a finite field, J. Lie Theory 12 (2002) 15

[30] Pari/Gp, version 2.1.6 (2005)

[31] V G Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter Co. (1994)

[32] S Wolfram, The Mathematica book, Wolfram Media, Champaign, IL (1999)