The connection of Morse function

Question

Suppose $M$ and $N$ are two manifolds, $f$ is a Morse function on $M$, $g$ is a Morse function on $N$, can you find a new manifold $P$ as the connection of $M$ and $N$ and a Morse function $h$ on the $P$, such that $h|_{M}$ is almost $f$, $h|_N$ is almost $g$ and the critical points of $h$ is the union the critical points of $f$ and $g$ (keep all the Morse indices same)?

Answer 1

Let's assume you actually mean that $P$ is the connected sum of $M$ and $N$. Then without adding new critical points this would not work - if it did the Euler characteristic would add up, but $E(P)=E(M)+E(N)+(1+(-1)^n)$ where $n$ is the dimension, so it would not work for even $n$.

What you could of course do, is a controlled modification. For example you could do connect sum near maximum of $f$ and minimum of $g$. The local level sets are spheres, identified to standard $S^{n-1}$ up to isotopy by the morse flow, and you can glue them to get the connected sum (for better results, glue the collar neighborhoods given by Morse flow). This has the rather obvious Morse function, which has as critical points exactly the critical points of $f$ and $g$ except the maximum and the minimum you used to connect sum. One advantage is that remaining critical points of $f$ form a subcomplex, and remaining critical points of $g$ form a quotient complex. This can be used to comute homology of $P$, for example.

Another option is to form a connect sum "near a regular point" of the morse flows of $f$ and $g$. Then you will get all critical points of $f$ and of $g$ as critical points, but you will also have 2 extra ones from the tube in the connect summ region, of index $1$ and $n-1$ (at least in the case of orientable connect sums).