I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked me a question that I couldn't answer. I don't know a lot about the Morse theory, so the question might actually be easy. I would really appreciate if you can help me, or at least give me a reference.
Suppose you are given an ordered set of signatures (i.e. number of $+$ and $-$ of the hessian) $\{(a_1,b_1),\dots,(a_r,b_r)\}$, that is supposed to be a set of critical points of some Morse function on a would be a $k$-manifold. The question is the following:
When there exists a manifold with a Morse function having a given set of signatures of critical points?
It is easy to see that the set of signatures must have signatures of the form $(k,0)$ and $(0,k)$, since any function on a compact manifold must have minimum and maximum.
Also, we can't start with, say, $(k-1,1)$, since you must start with the point of minimum, which must be of signature $(k,0)$.
Also, it is not true that we can always construct a manifold with given ordered set of signatures. For example, take the set $\{ (2,0) , (1,1), (0,2) \}$. Following the algorithm, first we attach a 0-cell, then we attach a 1-cell. Topologically it will be equivalent to a letter U made out of a tube (cylinder). But then you need two "caps" to make it into a closed compact thing, but we have only one critical point left.
I have no idea what are the conditions when we can actually construct a required manifold.
I've heard (but I am not sure if it is true) that if we have passed a sertain number of critical points in out reconstructing algorithm (maybe more than $r/2+1$), then there is unique way to finish the procedure to get a closed compact manifold. If this is correct, is it still true that we can always get a manifold having any set of the first $r/2+1$ signatures?
Thank you very much!