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I'm not sure if I am using the terms correctly. Suppose you have, for example, a Morse function $f : S^2 \rightarrow \mathbb{R}$ with 2 critical points of index 0, 1 critical point of index 1, and 1 critical point of index 2. Then it seems to me like we could homotope $f \sim g$ where $g$ is a Morse function on $S^2$ with just two critical points (I don't care whether this homotopy is via Morse functions or not, just as long as the final $g$ is a Morse function). I've been looking at Milnor's Morse Theory book but it doesn't seem like it discusses such procedures much. His book Lectures on the h-cobordism theorem seems to discuss a very similar procedure but for cobordisms and I'm not sure if I'm supposed to adapt that or what. I tried googling for cancellation of critical points and all I can find is this article, which looks like exactly the kind of thing I want, but not enough. If anyone can indicate a more complete portrait of this theory I will be very grateful.

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This is maybe not a full answer, but too long for a comment.

Any two Morse functions are homotopic through the obvious homotopy $t\mapsto t\,f+(1-t)g$. This homotopy will not be through Morse functions. This is easy to see for a closed manifold as the number of critical points cannot change through such a homotopy. The homotopies can be done through "generalized" Morse functions (but the formula above needs to be perturbed). Cerf theory studies this.

It is also clear that one needs conditions for cancellations (birth-death singularities) to be possible. It is not true that any two critical points of adjacent index can be cancelled. If this were true there existed a Morse function on the torus with no critical points. But the maximum and minimum of such a function are always critical so there always exist two critical points. A more advanced statement, the Morse inequalities, actually give a minimum number of four critical points on the torus.

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  • $\begingroup$ Yeah, I am aware you can't always cancel critical points. I see what you mean now, homotopies in this case are trivial. So I guess what I'm really looking for is good references on cancellation of critical points (when possible; what do we know about it?). $\endgroup$
    – Pedro
    May 21 '15 at 1:29
  • $\begingroup$ I've been looking at Matsumoto, Introduction to Morse Theory, and it's quite helpful; I plan to read what it has to say about it. $\endgroup$
    – Pedro
    May 21 '15 at 1:30

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