## Geometry & Topology

### Torus bundles not distinguished by TQFT invariants

Louis Funar

#### Abstract

We show that there exist arbitrarily large sets of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in $SL(2,ℤ)$ and its congruence quotients, the classification of SOL (polycyclic) 3–manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and by Ng and Schauenburg for the Drinfeld doubles of spherical fusion categories. In particular, we obtain non-isomorphic 3–manifold groups with the same pro-finite completions, answering a question of Long and Reid. On the other side we prove that two torus bundles over the circle with the same $U(1)$ and $SU(2)$ quantum invariants are (strongly) commensurable.

In the appendix (joint with Andrei Rapinchuk) we show that these examples have positive density in a suitable set of discriminants.

#### Article information

Source
Geom. Topol., Volume 17, Number 4 (2013), 2289-2344.

Dates
Revised: 23 April 2013
Accepted: 23 April 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732653

Digital Object Identifier
doi:10.2140/gt.2013.17.2289

Mathematical Reviews number (MathSciNet)
MR3109869

Zentralblatt MATH identifier
1278.57020

#### Citation

Funar, Louis. Torus bundles not distinguished by TQFT invariants. Geom. Topol. 17 (2013), no. 4, 2289--2344. doi:10.2140/gt.2013.17.2289. https://projecteuclid.org/euclid.gt/1513732653

#### References

• H Appelgate, H Onishi, Similarity problem over ${\rm SL}(n,\,{\bf Z}_{p})$, Proc. Amer. Math. Soc. 87 (1983) 233–238
• M Atiyah, The logarithm of the Dedekind $\eta$–function, Math. Ann. 278 (1987) 335–380
• B Bakalov, A Kirillov, Jr, Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001)
• P Bantay, The kernel of the modular representation and the Galois action in RCFT, Comm. Math. Phys. 233 (2003) 423–438
• T Barbot, Extensions de $\mathbb{Z}\oplus\mathbb{Z}$ par $\mathbb{Z}$, Mémoire DEA, ENS Lyon, UMPA (1990)
• J W Barrett, B W Westbury, Invariants of piecewise-linear $3$–manifolds, Trans. Amer. Math. Soc. 348 (1996) 3997–4022
• A I Borevich, I R Shafarevich, Number theory, Pure and Applied Mathematics 20, Academic Press, New York (1966)
• M R Bridson, S M Gersten, The optimal isoperimetric inequality for torus bundles over the circle, Quart. J. Math. Oxford Ser. 47 (1996) 1–23
• D Calegari, M H Freedman, K Walker, Positivity of the universal pairing in $3$ dimensions, J. Amer. Math. Soc. 23 (2010) 107–188
• J W S Cassels, Rational quadratic forms, London Mathematical Society Monographs 13, Academic Press, London (1978)
• J R Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973) 157–176
• H Cohn, A classical invitation to algebraic numbers and class fields, Springer, New York (1978) with an appendix by O Taussky-Todd
• A Coste, T Gannon, Congruence subgroups and rational conformal field theory
• F Deloup, Linking forms, reciprocity for Gauss sums and invariants of $3$–manifolds, Trans. Amer. Math. Soc. 351 (1999) 1895–1918
• F Deloup, An explicit construction of an abelian topological quantum field theory in dimension $3$, from: “Proceedings of the Pacific Institute for the Mathematical Sciences workshop “Invariants of three-manifolds””, (J Bryden, editor), Topology Appl. 127 (2003) 199–211
• F Deloup, C Gille, Abelian quantum invariants indeed classify linking pairings, from: “Proceedings of the conference Knots in Hellas '98, Vol. 2”, (C M Gordon, V F R Jones, L H Kauffman, S Lambropoulou, J H Przytycki, editors), J. Knot Theory Ramifications 10 (2001) 295–302
• J D Dixon, E W Formanek, J C Poland, L Ribes, Profinite completions and isomorphic finite quotients, J. Pure Appl. Algebra 23 (1982) 227–231
• C Dong, X Lin, S-H Ng, Congruence property in conformal field theory
• W Eholzer, On the classification of modular fusion algebras, Comm. Math. Phys. 172 (1995) 623–659
• P Etingof, On Vafa's theorem for tensor categories, Math. Res. Lett. 9 (2002) 651–657
• P Etingof, D Nikshych, V Ostrik, On fusion categories, Ann. of Math. 162 (2005) 581–642
• B Evans, L Moser, Solvable fundamental groups of compact $3$–manifolds, Trans. Amer. Math. Soc. 168 (1972) 189–210
• H G Feichtinger, M Hazewinkel, N Kaiblinger, E Matusiak, M Neuhauser, Metaplectic operators on $\mathbb C^n$, Q. J. Math. 59 (2008) 15–28
• É Fouvry, J Klüners, The parity of the period of the continued fraction of $\sqrt d$, Proc. Lond. Math. Soc. 101 (2010) 337–391
• D S Freed, F Quinn, Chern–Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993) 435–472
• M H Freedman, V Krushkal, On the asymptotics of quantum ${\rm SU}(2)$ representations of mapping class groups, Forum Math. 18 (2006) 293–304
• L Funar, Représentations du groupe symplectique et variétés de dimension $3$, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 1067–1072
• L Funar, Theta functions, root systems and $3$–manifold invariants, J. Geom. Phys. 17 (1995) 261–282
• L Funar, Some abelian invariants of $3$–manifolds, Rev. Roumaine Math. Pures Appl. 45 (2000) 825–861
• L Funar, T Kohno, On Burau representations at roots of unity, to appear in Geometriae Dedicata (2013)
• T Gannon, Modular data: the algebraic combinatorics of conformal field theory, J. Algebraic Combin. 22 (2005) 211–250
• C F Gauss, Disquisitiones arithmeticae, Yale University Press, New Haven, Conn. (1966)
• E Ghys, V Sergiescu, Stabilité et conjugaison différentiable pour certains feuilletages, Topology 19 (1980) 179–197
• P M Gilmer, Congruence and quantum invariants of $3$–manifolds, Algebr. Geom. Topol. 7 (2007) 1767–1790
• T Gocho, The topological invariant of three-manifolds based on the ${\rm U}(1)$ gauge theory, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39 (1992) 169–184
• F J Grunewald, P F Pickel, D Segal, Polycyclic groups with isomorphic finite quotients, Ann. of Math. 111 (1980) 155–195
• A J Hahn, O T O'Meara, The classical groups and $K$–theory, Grundl. Math. Wissen. 291, Springer, Berlin (1989)
• H Halberstam, A proof of Chen's theorem, from: “Journées Arithmétiques de Bordeaux”, Astérisque 24–25, Soc. Math. France, Paris (1975) 281–293
• L C Jeffrey, Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Comm. Math. Phys. 147 (1992) 563–604
• N Kaiblinger, M Neuhauser, Metaplectic operators for finite abelian groups and $\mathbb R^d$, Indag. Math. 20 (2009) 233–246
• J Kania-Bartoszynska, Examples of different $3$–manifolds with the same invariants of Witten and Reshetikhin–Turaev, Topology 32 (1993) 47–54
• R Kirby, P Melvin, Dedekind sums, $\mu$–invariants and the signature cocycle, Math. Ann. 299 (1994) 231–267
• M Lackenby, Fox's congruence classes and the quantum-${\rm SU}(2)$ invariants of links in $3$–manifolds, Comment. Math. Helv. 71 (1996) 664–677
• M Larsen, Z Wang, Density of the $\mathrm{SO}(3)$ TQFT representation of mapping class groups, Comm. Math. Phys. 260 (2005) 641–658
• W B R Lickorish, Distinct $3$–manifolds with all ${\rm SU}(2)_q$ invariants the same, Proc. Amer. Math. Soc. 117 (1993) 285–292
• D D Long, A W Reid, Grothendieck's problem for $3$–manifold groups, Groups Geom. Dyn. 5 (2011) 479–499
• G Masbaum, On representations of mapping class groups in integral TQFT, Oberwolfach Reports 5 (2008) 1202–1205
• W Meyer, Die Signatur von Flächenbündeln, Math. Ann. 201 (1973) 239–264
• R A Mollin, Fundamental number theory with applications, 2nd edition, Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton (2008)
• M Müger, From subfactors to categories and topology, II: the quantum double of tensor categories and subfactors, J. Pure Appl. Algebra 180 (2003) 159–219
• H Murakami, T Ohtsuki, M Okada, Invariants of three-manifolds derived from linking matrices of framed links, Osaka J. Math. 29 (1992) 545–572
• M Newman, Integral matrices, Pure and Applied Mathematics 45, Academic Press, New York (1972)
• S-H Ng, P Schauenburg, Congruence subgroups and generalized Frobenius–Schur indicators, Comm. Math. Phys. 300 (2010) 1–46
• N Nikolov, D Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, Ann. of Math. 165 (2007) 171–238
• I Niven, H S Zuckerman, H L Montgomery, An introduction to the theory of numbers, 5th edition, John Wiley & Sons, New York (1991)
• P F Pickel, Finitely generated nilpotent groups with isomorphic finite quotients, Trans. Amer. Math. Soc. 160 (1971) 327–341
• V Platonov, On the genus problem in arithmetic groups, Dokl. Akad. Nauk SSR 200 (1971) 793–796 In Russian; translated in Soviet. Math. Dokl. 12 (1975), 1503–1507
• V Platonov, A Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press, Boston, MA (1994)
• G Prasad, A Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. IHÉS (2009) 113–184
• G Prasad, A Rapinchuk, Number-theoretic techniques in the theory of Lie groups and differential geometry, from: “Fourth International Congress of Chinese Mathematicians”, (L Ji, K Liu, L Yang, S-T Yau, editors), AMS/IP Stud. Adv. Math. 48, Amer. Math. Soc. (2010) 231–250
• H Rademacher, E Grosswald, Dedekind sums, The Carus Mathematical Monographs 16, The Mathematical Association of America, Washington, DC (1972)
• A Rapinchuk, Platonov's conjecture on genus in arithmetic groups, Dokl. Akad. Nauk BSSR 25 (1981) 101–104, 187 In Russian; translated in Amer. Math. Soc. Translations–Series 2 128 (1986), 117–122
• L Redei, H Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, J. Reine Angew. Math. 170 (1933) 69–74
• N Reshetikhin, V G Turaev, Invariants of $3$–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547–597
• H-E Richert, Lectures on sieve methods, Lectures on Mathematics and Physics 55, Bombay: Tata Institute of Fundamental Research (1976)
• G Sabbagh, J S Wilson, Polycyclic groups, finite images, and elementary equivalence, Arch. Math. $($Basel$)$ 57 (1991) 221–227
• P Sarnak, Reciprocal geodesics, from: “Analytic number theory”, (W Duke, Y Tschinkel, editors), Clay Math. Proc. 7, Amer. Math. Soc. (2007) 217–237
• P F Stebe, Conjugacy separability of groups of integer matrices, Proc. Amer. Math. Soc. 32 (1972) 1–7
• P Stevenhagen, The number of real quadratic fields having units of negative norm, Experiment. Math. 2 (1993) 121–136
• T Tatuzawa, On a theorem of Siegel, Jap. J. Math. 21 (1951) 163–178
• C Traina, A note on trace equivalence in ${\rm PSL}(2,{\bf Z})$, Rend. Istit. Mat. Univ. Trieste 26 (1994) 233–237
• V G Turaev, Quantum invariants of knots and $3$–manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter & Co, Berlin (1994)
• V G Turaev, A Virelizier, On two approaches to $3$–dimensional TQFTs
• V G Turaev, O Y Viro, State sum invariants of $3$–manifolds and quantum $6j$–symbols, Topology 31 (1992) 865–902
• C Vafa, Toward classification of conformal theories, Phys. Lett. B 206 (1988) 421–426
• F Xu, Some computations in the cyclic permutations of completely rational nets, Comm. Math. Phys. 267 (2006) 757–782