Defining and classifying TQFTs via surgery

Abstract

We give a presentation of the $n$-dimensional oriented cobordism category ${\text{Cob}}_n$ with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor $F$ from the category of smooth oriented manifolds and diffeomorphisms to an arbitrary category $C$, and morphisms induced by surgeries along framed spheres, we obtain a necessary and sufficient set of relations these have to satisfy to extend to a functor from ${\text{Cob}}_n$ to $C$. If $C$ is symmetric and monoidal, then we also characterize when the extension is a TQFT. This framework is well-suited to defining natural cobordism maps in Heegaard Floer homology. It also allows us to give a short proof of the classical correspondence between (1+1)-dimensional TQFTs and commutative Frobenius algebras. Finally, we use it to classify (2+1)-dimensional TQFTs in terms of J-algebras, a new algebraic structure that consists of a split graded involutive nearly Frobenius algebra endowed with a certain mapping class group representation. This solves a long-standing open problem. As a corollary, we obtain a structure theorem for (2+1)-dimensional TQFTs that assign a vector space of the same dimension to every connected surface. We also note that there are $2^{2^{\omega }}$ nonequivalent lax monoidal TQFTs over $\mathbb{C}$ that do not extend to (1+1+1)-dimensional ones.

68 pages, 4 figures, to appear in Quantum Topology

References in corpus (3)

• Modular categories as representations of the 3-dimensional bordism 2-category
• Quasistrict symmetric monoidal 2-categories via wire diagrams
• Extended 3-dimensional bordism as the theory of modular objects

Cited by in corpus (9)

• Modular categories as representations of the 3-dimensional bordism 2-category
• Link cobordisms and functoriality in link Floer homology
• Graph cobordisms and Heegaard Floer homology
• Introductory lectures on topological quantum field theory
• Tangle Floer homology and cobordisms between tangles
• Unoriented Cobordism Maps on Link Floer Homology
• Higher categories of push-pull spans, I: Construction and applications
• The Cyclic and Modular Microcosm Principle in Quantum Topology
• On the Faithfulness of 1-dimensional Topological Quantum Field Theories