Defining and classifying TQFTs via surgery

TL;DR

This paper provides a comprehensive algebraic framework for defining and classifying topological quantum field theories (TQFTs) via surgery presentations, solving longstanding classification problems in low-dimensional topology.

Contribution

It introduces a new presentation of the cobordism category using surgeries, characterizes when functors extend to TQFTs, and classifies (2+1)-dimensional TQFTs via J-algebras, addressing open problems.

Findings

Classifies (2+1)-dimensional TQFTs using J-algebras.

Provides a necessary and sufficient set of relations for functors to extend to cobordism categories.

Shows the existence of many non-extendable lax monoidal TQFTs over a3.

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.