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First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$.

For simplicity, let's restrict for now to the specific case of $M^4 = B^4\cup_\varphi\text{2-handle}$, where the 2-handle is a copy of $D^2\times D^2$.

The attaching map $\varphi:\partial D^2\times D^2\to\partial B^4$ is determined by:

  • an isotopy class of $\left.\varphi\right\rvert_{S^1\times\{0\}}$, i.e. a knot $K:\partial D^2\times D^2\supset S^1\times\{0\}\to\mathbb{S}^3=\partial B^4$,
  • and a framing of the normal bundle of $\varphi(S^1\times\{0\})$, i.e. an identification of $\nu(K)$ (a tubular neighbourhood of $K$) with $S^1\times D^2$.

My question is, if I start with such a manifold as above, and a Morse function on it, can I somehow recover the (or any) attaching information (the framing, or the attaching region, or all of it...) of the handle(s)?

Practically speaking, let's say we have a triangulation of $M$, and a discrete Morse function on it. This allows me to "see" the handle decomposition at the level of "there is a 2-handle" (e.g. the Morse function gives that, say, "triangle 5" is a critical cell), but I am wondering if it is possible to extract any information about the attaching map? Maybe the attaching circle can be recovered by looking at the edges of the critical triangle... But what about the framing? Maybe to see anything about the framing we need to drill out a neighbourhood of this critical triangle... but I wonder if anything can be extracted just from the Morse function alone? The literature would seem to indicate that either we can't, or nobody has described how to do so if it is possible.

Any thoughts/information is appreciated.

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  • $\begingroup$ People do these things in the literature. Perhaps you are reading the wrong sources. The knots and their associated vector fields can be derived from the critical point data and flows of the associated vector fields. It's spelled out quite explicitly in books like Kosinski's Differential Topology. Often high-level texts won't get into these sorts of details as they're too computational, visual or geometric (choose appropriate language). $\endgroup$
    – Ryan Budney
    Commented Feb 17, 2023 at 2:57
  • $\begingroup$ @RyanBudney I probably should have just emailed you to begin with in hindsight, you're probably the perfect person to have asked! My apologies if it's trivial to see this, but would you be willing to elaborate on this a bit, I don't quite see how you would recover the knot+framing just from the critical point..? To further clarify (if it makes a difference), my discrete morse functions here are in the form of acyclic matchings on the Hasse diagram of the triangulation (and I'm working with generalised triangulations (Regina)). $\endgroup$
    – rab
    Commented Feb 17, 2023 at 3:22

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Near a critical point $p$ of a genuine Morse function you have the local model

$$f(x) = f(p) + x_1^2 + \cdots + x_i^2 - x_{i+1}^2 - \cdots - x_n^2$$

Just below the height of $p$, your attaching sphere is the unit sphere in the plane $\{0\} \times \mathbb R^{n-i}$ corresponding to all the $x_j^2$'s with minus signs in the above formula.

So you apply the flow of the negative gradient function to that sphere together with its standard framing. i.e. we are taking the flow not just on the manifold, but on the tangent bundle (i.e. the derivative of the manifold flow).

Once you flow that sphere down to whatever level you like, you have flowed the framing vector field as well. That's your attachment framing.

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