# References on cancellation of critical points

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.

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.