Torus bundles not distinguished by TQFT invariants. With an appendix by Louis Funar and Andrei Rapinchuk
type: Item
instance of: scholarly article
title: Torus bundles not distinguished by TQFT invariants. With an appendix by Louis Funar and Andrei Rapinchuk
zbMATH Open document ID: 1278.57020
published in: Geometry \& Topology
publication date: 2013-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.0509
reviewed by: Daniel Moskovich
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 6206250
zbMATH Keywords: mapping class group
zbMATH Keywords: torus bundle
zbMATH Keywords: modular tensor category
zbMATH Keywords: congruence subgroup
zbMATH Keywords: SL\((2, \mathbb Z)\)
zbMATH Keywords: conjugacy problem
zbMATH Keywords: Pell equations
zbMATH Keywords: TQFT
zbMATH Keywords: Turaev-Viro invariants
zbMATH Keywords: Reshetikhin-Turaev invariants
zbMATH Keywords: MOO invariant
zbMATH Keywords: SOL manifolds
zbMATH Keywords: commensurable groups
zbMATH Keywords: profinite groups
zbMATH Keywords: congruent manifolds
author: Louis Funar
MaRDI profile type: Publication
OpenAlex ID: W3106291025
arXiv ID: 1101.0509
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Torus bundles not distinguished by TQFT invariants. With an appendix by Louis Funar and Andrei Rapinchuk (DOI)
DOI: 10.2140/GT.2013.17.2289

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review text: 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 ha…