# Defining and classifying TQFTs via surgery

Juhasz, A

2018

QUANTUM TOPOLOGY

## Last Updated:

2020-01-12T22:38:11.243+00:00

2

9

10.4171/QT/108

229-321

## abstract:

© European Mathematical Society. We give a presentation of the n-dimensional oriented cobordism category Cobn 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 Cobn 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 22ω nonequivalent lax monoidal TQFTs over C that do not extend to (1+1+1)-dimensional ones.

729215

Submitted

Journal Article