Today I’ll just prove that a Morse function will give a handle decomposition of a closed manifold. Let’s use all the notation already set up (meaning critical points, values, attaching maps, dimension, Morse function, gradient-like vector field, etc).
We just induct on the subscripts of critical points. We’ve already done the base case (it is a min and hence a 0-handle from here). So we just need to show that if is a handlebody for , then is a handlebody with the appropriate handle attached.
So we’ve assumed that we have some decomposition . We also know that we attach a handle of index when crossing , so we do have a diffeo to a manifold with a -handle attached with attaching map .
Note that contains no critical values, so by flowing along we get a diffeo . Let be this diffeo.
So by inductive hypothesis, , so we can assume actually maps from the handlebody to . Now by composing we get our actual attaching map (note that before now the handle was attached to and not the handlebody itself).
i.e. . So let , and we get that , so we are done.
So I sort of dragged on longer than probably necessary there, since there was essentially nothing new. It was just being pedantic about the diffeo of the manifold and the handlebody.
There are some subtleties that should be pointed out, though. The index of the critical point did determine the index of the handle, and we went in “ascending” order. The other much more important and also more subtle point is that the choice of gradient-like vector field was how we constructed the attaching map. So even the same Morse function with a different choice of gradient-like vector field could actually give a “different” handle decomposition when considering attaching maps as part of the data.