Restricted (2+1)-TQFTs supported by thickened and solid tori

TL;DR

This paper constructs specific 3-dimensional topological quantum field theories (TQFTs) supported on thickened and solid tori, capable of distinguishing certain 3-manifolds that previous quantum invariants could not differentiate.

Contribution

It introduces a new subcategory of 3-cobordisms involving tori and defines TQFTs that can distinguish complex 3-manifolds, advancing understanding of (2+1)-TQFTs.

Findings

TQFTs distinguish pairs of torus bundles and lens spaces

Subcategory includes disjoint unions of tori and simple cobordisms

Advances the study of (2+1)-TQFTs and their distinguishing power

Abstract

A faithful $(1+1)$ TQFT has recently been constructed, but the existence of a faithful $(2+1)$ TQFT remains an open question, that subsumes the hard problem of linearity of mapping class groups of surfaces. To circumvent the latter problem we construct a subcategory of the category of 3-cobordisms, containing disjoint unions of tori and simplest cobordisms between them. On this we define TQFTs that are able to distinguish pairs of torus bundles and lens spaces, previously shown not to be distinguishable by quantum invariants.