Defining and classifying TQFTs via surgery

We give a presentation of the n -dimensional oriented cobordism category 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 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^{2ω} nonequivalent lax monoidal TQFTs over C that do not extend to (1+1+1)-dimensional ones.