Geometry & Topology

Torus bundles not distinguished by TQFT invariants

Louis Funar

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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,) 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

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

Received: 3 March 2011
Revised: 23 April 2013
Accepted: 23 April 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups 57M07: Topological methods in group theory
Secondary: 20F38: Other groups related to topology or analysis 57N05: Topology of $E^2$ , 2-manifolds

mapping class group torus bundle modular tensor category congruence subgroup $\mathrm{SL}(2,\mathbb{Z})$ conjugacy problem Pell equation rational conformal field theory


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

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