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

2025/05/27 by Dušan Ɖorđević, Đorđević, Dušan, Danica Kosanović +5
Engineering · Materials Science · #18A22 #18M05 #57K16 #57K30 #57R56 #Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT) #Silicone and Siloxane Chemistry #Structural Behavior of Reinforced Concrete

paper · pdf · doi:10.48550/arxiv.2505.21373

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The Dehn Twist Action for Quantum Representations of Mapping Class Groups
• A diagrammatic presentation of the category 3Cob
• On mapping class groups and their TQFT representations
• A faithful 2-dimensional TQFT
• Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants
• Representations of the braid group B₃ and of SL(2,Z)
• Quantum Invariants of Knots and 3-Manifolds
• Invariants of 3-manifolds via link polynomials and quantum groups
• Curves on 2-manifolds and isotopies

Related

• Non semi-simple TQFTs from unrolled quantum sl(2)
• A note on TQFTs for orientable 2-dimensional cobordisms
• Hennings TQFTs for Cobordisms Decorated With Cohomology Classes
• Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants
• 3-cobordisms with their rational homology on the boundary