Torus bundles not distinguished by TQFT invariants. With an appendix by Louis Funar and Andrei Rapinchuk
The two main questions for any quantum topological invariant are ``What is its topological interpretation?'', and ``Does it separate the topological objects on which it is defined?''.NEWLINENEWLINEThis breakthrough paper exhibits arbitrarily large sets of non-homeomorphic closed oriented \(3\)--manifolds which are not separated by any Reshetikhin--Turaev invariant coming from a modular tensor category \(\mathcal{A}\) (Corollary 1.1). This answers in the negative a well-known question of \textit{V. G. Turaev} [Quantum invariants of knots and \(3\)--manifolds. de Gruyter Studies in Mathematics 18. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)], whether such invariants classify closed oriented \(3\)--manifolds. Indeed, counterexamples may be chosen to be prime (Theorem 1.3). Conjecturally, this result implies that this family of \(3\)--manifolds, which are SOL torus bundles, are not separated by any \(2\)--functor for which the value of circles is a modular tensor category \(\mathcal{A}\) with monoidal structure induced from the pair of pants and braiding coming from the half-twist on the pair of pants.NEWLINENEWLINEThe main technical tool in the paper is the congruence subgroup property for representations of \(SL(2,\mathbb{Z})\) associated to Drinfel'd doubles of spherical fusion categories [\textit{S.-H. Ng} and \textit{P. Schauenburg}, Commun. Math. Phys. 300, No. 1, 1--46 (2010; Zbl 1206.18007)]. Combined with the classification of SOL 3-manifolds and some arithmetic, the problem reduces to an arithmetic question about integral matrices.NEWLINENEWLINEThese results lead towards a conjectured topological interpretation of Turaev--Viro invariants over \(\mathbb{C}\) of fibered \(3\)--manifolds, namely that the information that they detect is the profinite fundamental group of the manifold (and a little bit more---see Remark 1.4). Also, Conjecture 1.1 states that these invariants coincide if and only if the fundamental groups are commensurable, meaning that they have all isomorphic finite index subgroups. This in particular implies that Turaev-Viro invariants should detect the geometric type of a \(3\)--manifold. For torus bundles over the circle, commensurability is detected by \(U(1)\) and \(SU(2)\) quantum invariants.NEWLINENEWLINEThe counterexamples in the paper also provide examples of non-isomorphic \(3\)--manifold groups with the same profinite completion, answering a question of \textit{D. D. Long} and \textit{A. W. Reid} [Groups Geom. Dyn. 5, No. 2, 479--499 (2011; Zbl 1250.57027). ]
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