Morse Functions Exist

The astute reader at this point may be getting a little anxious that despite the fact that I found Morse function in two easy low dimensional cases, my eventual goal of saying very general things about manifolds by using Morse functions is going to rely on the fact that they exist.

If these thing are really as powerful as I have been making them out to be, then it would seem that there probably isn’t an abundance of them. But surprisingly, it turns out that basically every smooth function is Morse.

Let $M^n$ be a closed manifold, and $g:M\to \mathbb{R}$ be a smooth function. Then there is a Morse function $f:M\to\mathbb{R}$ arbitrarily close to $g$.

Recall Sard’s Theorem (I’m assuming some familiarity with it, which is probably not a good idea): The set of critical values of a smooth map $f: U\to \mathbb{R}^n$ has measure zero in $\mathbb{R}^n$.

Now we’ll first need a lemma. Let $U\subset \mathbb{R}^n$ be an open set and $f:U\to\mathbb{R}$ a smooth function. Then there are real numbers $\{a_k\}$ such that $f(x_1, \ldots, x_n)-(a_1x_1+a_2x_2+\cdots + a_nx_n)$ is a Morse function on $U$. We can also choose $\{a_k\}$ to be arbitrarily small in absolute value.

Let $p\in U$ be a critical point of $f$. Define $h=Jac(f)^T$ (a smooth map $h:U\to\mathbb{R}^n$). Then $Jac(h)\Big|_p$ is the Hessian $H_f(p)$. Thus, p is a critical point of $h$ iff $det(H_f(p))=0$.

By Sard’s Theorem, we can choose $a=(a_1, \ldots , a_n)\in\mathbb{R}^n$ where each $a_k$ have arbitrarily small absolute value such that $a$ is not a critical value of $h$.

The claim is that $\overline{f}(x_1, \ldots , x_n)=f(x_1, \ldots, x_n)-(a_1x_1+\cdots + a_nx_n)$ is a Morse function on U.

Well, if $p$ is a critical point of $\overline{f}$, then since $\frac{\partial \overline{f}}{\partial x_i}\Big|_p=\frac{\partial f}{\partial x_i}\Big|_p - a_i=0$, by the definition of h, we get $h(p)=a$.

But we chose $a$ to not be a critical value of h. Thus, p is not a critical point of h. So as noted, $det(H_f(p))\neq 0$. But $H_f(p)=H_{\overline{f}}(p)$, so $p$ is a non-degenerate critical point. Since p was an arbitrary critical point, all critical points are non-degenerate and hence $\overline{f}$ is Morse, completing the proof of the Lemma.

We also need another Lemma. Let $K\subset M$ be a compact subset. Then if $g:M\to\mathbb{R}$ has no degenerate critical points in $K$, then we can choose $\varepsilon >0$ small enough so that any $C^2$ approximation of $g$ also has no degenerate critical points in $K$.

Since our manifold is closed, it is compact. So we can choose a finite subcover of coordinate charts, and compactly refine it (I’ll do this construction if someone asks in the comments), so that $\{U_i\}_1^m$ cover $M$ and there are compact sets $K_i\subset U_i$ such that $\cup K_i=M$.

But with this, we can look at any of the $U_k$, and in these coordinates, $g$ has no degenerate critical points in $K\cap K_k$ (alright, that was probably a poor choice of notation) iff $\displaystyle\Big|\frac{\partial g}{\partial x_1}\Big|+\cdots + \Big|\frac{\partial g}{\partial x_n}\Big|+\Big| det(H_g)\Big|>0$ for every point in $K\cap K_k$.

But for a small enough $\varepsilon$ we can definitely still make that inequality hold for any $C^2$ approximation. Thus we have proved the lemma.

Now let’s do the actual existence proof. Take the $U_i, K_i$ as before. We will inductively build our $C^2$ approximations on $C_l=K_1\cup \cdots \cup K_l$. Our base step is to build $f_0$ on $C_0=\emptyset$, so we’re done.

For our inductive hypothesis, suppose we have $f_{l-1}:M\to\mathbb{R}$ having no degenerate critical points in $C_{l-1}$.

Let’s work with the coordinate neighborhood $U_l$ with coordinates $(x_i)$. By the first lemma, there are arbitrarily small numbers $\{a_i\}$ so that $f_{l-1}(x_1, \ldots , x_n)-(a_1x_1+\cdots + a_nx_n)$ is Morse on $U_l$. But note, we only have a definition on $U_l$ and we need one everywhere.

Let $\psi$ be a bump function that is 1 on $K_l$ and supported in $V$, where $K_l\subset V\subset U_l$.

Define $f_l=\begin{cases} f_{l-1}-\psi\cdot (a_1x_1+\cdots a_nx_m) & in \ U_l \\ f_{l-1} & outside \ V\end{cases}$.

(So I have this same cases problem again, just ignore the “line break” symbol, it is actually readable this time).

This gives us a nice well-defined function on all of $M$ (just need to check the overlaps). Also $f_l$ is our first lemma function on $K_l$, so it is Morse on $K_l$ and hence has no degenerate critical points there.

Since $0\leq \psi \leq 1$ (and we’re on a compact set), we can make $\{a_i\}$ small enough so that $f_l$ is an arbitrarily close $C^2$ approximation of $f_{l-1}$ (I won’t do this since it is fairly long and tedious, but quite straightforward for the reasons I gave).

But now by the second lemma, since $f_{l-1}$ has no degenerate critical points in $C_{l-1}$, we have that $f_l$ has no degenerate critical points in $C_{l-1}$ either. We already checked on $K_l$, and thus there are no deg. critical points on $C_{l-1}\cup K_l=C_l$.

Thus inductively we can get a Morse function on all of $M$ that is $C^2$-close to our original smooth function.