So, in another thread Scott C. questioned why I would say that TQFT’s produce all numerical invariants of n-manifolds. There is a good reason for this, though maybe not good enough.
The reason is that there is a construction for starting with a numerical invariant, and extending it to a TQFT:
- Start with a numerical invariant of n-manifolds.
- Now consider your favorite n-1-manifold X.
- Consider the vector space spanned by n-manifolds with isomorphisms of their boundary to X.
- The space has an inner product, given by gluing two n-manifolds along X, and summing their invariants. Quotient by the kernel of this and call this .
Congratulations. You’ve got your TQFT. This construction is a little weird; in particular, it doesn’t seem to have to be monoidal, which is bad. Still, it tells you TQFT’s are in some sense easier to construct than you probably thought.
Presumably, one can extend this further by giving an n-2-manifold the category whose objects are formal sums of manifolds bounding it, with morphism spaces given by gluing, etc. Anyways, this is what I had in mind when writing the infamous sentence in question.