Defining and classifying tqfts via surgery

Author: 

Juhász, A

Publication Date: 

1 January 2018

Journal: 

Quantum Topology

Last Updated: 

2019-04-27T04:23:11.92+01:00

Issue: 

2

Volume: 

9

DOI: 

10.4171/QT/108

page: 

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.

Symplectic id: 

729215

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article