We describe a framework for defining and classifying TQFTs via
surgery. Given a functor
from the category of smooth manifolds and diffeomorphisms to
finite-dimensional vector spaces,
and maps induced by surgery along framed spheres, we give a set of axioms
that allows one to assemble functorial coboridsm maps.
Using this, we can reprove the correspondence between (1+1)-dimensional
TQFTs and commutative Frobenius algebras,
and classify (2+1)-dimensional TQFTs in terms of a new structure, namely
split graded involutive nearly Frobenius algebras
endowed with a certain mapping class group representation. The latter has
not appeared in the literature even in conjectural form.
This framework is also well-suited to defining natural cobordism maps in
Heegaard Floer homology.