# Is there a reasonable definition of TQFTs for n-cobordisms with connected inputs/outputs?

A question that's been on my mind for a while is whether any precise statement to the effect of "Heegaard Floer homology is a TQFT," for some reasonable definition of TQFT, can be made.

Of course, a lot of effort is being spent right now on HF as an extended TQFT (e.g. the bordered theory of Lipshitz/Ozsvath/Thurston and, more recently, the cornered theory of Douglas/Lipshitz/Manolescu). But right now I'm just wondering about the 3+1 structure.

The issue in 3+1 dimensions (leaving aside the mixed invariants and how to derive them from a TQFT framework) is that only cobordisms between connected 3-manifolds induce maps on HF. This is, in some sense, a fundamental feature of the theory, since the induced maps for closed 4-manifolds are zero. This was discussed in the MO question Seiberg-Witten theory on 4-manifolds with boundary.

So, what if one were to make this feature into a definition? "Some variant of TQFT" := a functor which only allows these cobordisms with connected inputs and outputs? Does this correspond to some definition that's already out there? Is it a reasonable thing to consider in the framework of, e.g., Lurie's classification of fully extended TQFTs? Or is there some other definition which could be used instead, more amenable to this framework?

I'm putting a "reference-request" tag on this question, because answering it as stated probably would consist of pointing out a relevant paper or two, but I'd be interested more generally in anything that continues the discussion from the MO question I linked above.

• Then when you take Hochschild homology of this algebra, it gives you HFK of $pt \times S^1$ inside $F \times S^1$ (if $F$ is the surface), rather than HF of $F \times S^1$. In 4 dimensions, if you have a closed 3-manifold $Y$ and look at $Y \times S^1$, you could remove two balls and get a cobordism from $S^3$ to $S^3$. Then you might get confused, since this cobordism induces the zero map on HF, whereas in the usual TQFT picture it should be multiplication by the dimension of HF(Y). But in this perspective, maybe the more natural thing is remove the nbhd of two points from $Y$, rather (...) Oct 27, 2013 at 19:22
• than $Y \times S^1$. Then, crossing with $S^1$, you get a cobordism from $S^2 \times S^1$ to itself, which can recover $Y \times S^1$ by gluing in two copies of $B^3 \times S^1$. The map induced on HF of $S^2 \times S^1$ is still zero, but by the previous philosophy, maybe one should consider ($pt \times S^1$ inside $S^2 \times S^1$) as a more natural thing here than just $S^2 \times S^1$. I'm not sure off the top of my head whether the induced map on HFK of $pt \times S^1$ is still zero. At least it's not equivalent to a closed surface invariant, because the $pt \times S^1$s are essential. Oct 27, 2013 at 19:27