[Paper Review] Three-Dimensional 2-Framed TQFTs and Surgery

Stephen Sawin|ArXiv.org|Dec 8, 1999

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper constructs a 2-framed three-dimensional topological quantum field theory (TQFT) using surgery presentations of 3-manifolds and a skeletal category of 2-framed cobordisms. It provides a finite set of algebraic data and relations—expressed in the language of surgery—that completely classify such TQFTs, offering a complete algebraic characterization of 2-framed 3D TQFTs via surgery calculus.

ABSTRACT

The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are expressed in the language of surgery.

Motivation & Objective

• To define and formalize the notion of 2-framed 3-manifolds and their cobordisms.
• To construct a 2-framed 3-dimensional topological quantum field theory (TQFT) using a category of 2-framed cobordisms.
• To provide a finite, algebraic presentation of such TQFTs using surgery calculus.
• To show that the TQFT structure is completely determined by a small set of data and relations in the surgery language.

Proposed method

• The paper introduces the category of 2-framed cobordisms, where objects are 2-framed 3-manifolds and morphisms are cobordisms with 2-framing data.
• It employs skeletonization techniques to reduce the category of 2-framed cobordisms to a finitely presented algebraic structure.
• The construction uses surgery presentations of 3-manifolds, expressing the TQFT in terms of framed links and surgery relations.
• The TQFT is defined by assigning algebraic data (e.g., vector spaces, linear maps) to surgery presentations, satisfying relations derived from Kirby calculus.
• The core relations are derived from the structure of the 2-framed cobordism category and encoded in the language of surgery moves.
• The resulting TQFT is shown to be fully determined by a finite set of generators and relations, making it amenable to explicit computation.

Experimental results

Research questions

• RQ1 How can a 2-framed 3-dimensional TQFT be algebraically classified using surgery presentations of 3-manifolds?
• RQ2 What is the minimal set of algebraic data and relations required to define a 2-framed 3D TQFT?
• RQ3 How does the category of 2-framed cobordisms relate to the surgery calculus of 3-manifolds?
• RQ4 Can a 2-framed TQFT be fully reconstructed from a finite presentation of its data and relations in the surgery language?
• RQ5 What role does skeletonization play in simplifying the structure of 2-framed TQFTs?

Key findings

• The paper successfully defines the category of 2-framed cobordisms, providing a rigorous foundation for 2-framed 3D TQFTs.
• It constructs a 2-framed 3D TQFT by assigning algebraic data to surgery presentations of 3-manifolds.
• The TQFT is completely determined by a finite set of generators and relations, expressed in the language of surgery.
• The construction shows that 2-framed TQFTs are fully classified by their behavior under surgery moves, including the Kirby moves.
• The resulting TQFT structure is equivalent to a representation of the 2-framed cobordism category, with all invariants determined by the algebraic data.
• The paper establishes that the TQFT is well-defined and consistent under the standard surgery calculus, validating its use in topological invariants.