# Defining and classifying TQFTs via surgery

### András Juhász

University of Oxford, UK

## Abstract

We give a presentation of the $n$-dimensional oriented cobordism category $\mathbf {Cob}_n$ 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 sucient set of relations these have to satisfy to extend to a functor from $\mathbf {Cob}_n$ 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.

## Cite this article

András Juhász, Defining and classifying TQFTs via surgery. Quantum Topol. 9 (2018), no. 2, pp. 229–321

DOI 10.4171/QT/108