Torus bundles not distinguished by TQFT invariants

TL;DR

This paper demonstrates that many non-homeomorphic SOL torus bundles share the same TQFT invariants, revealing limitations of these invariants in distinguishing certain 3-manifolds, and explores their arithmetic and group-theoretic properties.

Contribution

It constructs infinite pairs of non-homeomorphic SOL torus bundles with identical quantum invariants and proves they are strongly commensurable, highlighting the invariants' limitations.

Findings

Existence of infinitely many non-homeomorphic torus bundles with identical TQFT invariants.

Quantum invariants do not distinguish non-homeomorphic SOL torus bundles.

Such bundles are shown to be strongly commensurable.

Abstract

We show that there exist infinitely many pairs 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 lastly by Ng and Schauenburg for the Drinfeld doubles of spherical fusion categories. On the other side we prove that two torus bundles over the circle with the same quantum invariants are (strongly) commensurable. The examples above show that this is the best that it could be expected.