Existence of a proper Morse function

Question

Given a manifold $M$, I know there exists a proper function $f: M \to \mathbb R$ (using the usual partition of unity argument) and a Morse function $g: M \to \mathbb R$ (genericity of Morse functions). However, I am not sure how to prove the existence of a proper Morse function.

My initial idea is that, if $f$ has isolated critical points, then we can perturb $f$ locally whenever the critical points are degenerate so that the resulting $f$ is Morse. Moreover local perturbations by some bounded quantity preserves properness. I haven't been able to prove that proper functions have isolated critical points though, so I was wondering if I was on the right track.

Answer 1

Initially I was looking for something that used a little less machinery, but the problem is pretty easy once we assume Whitney embedding.

Let $f: M \to \mathbb R^m$ be an embedding, which we can find for $m$ sufficiently large, so now we just think of $M \subseteq \mathbb R^m$. We can translate $M$ so that it does not hit the origin, in which case the norm map $g: x \mapsto |x|$ is smooth on $M$ and proper. By theorem in Guillemin and Pollack, the map \begin{equation*} g_a = g + a \cdot x \end{equation*} is Morse for almost ever $a \in \mathbb R^m$. Choose some $a$ with sufficiently small norm, say $|a|<1/2$, such that $g_a$ is Morse, then $|g_a (x)| \geq |x|/2$, so $g_a$ is also proper.

I guess the question still stands, is there a way of doing this without Whitney embedding?

Answer 2

I will write a very fancy and unnecessarily complicated proof. However, it is my expectation that this proof will be illuminating. Similar ideas are used in other contexts, such as Morse theory on infinite-dimensional manifolds. Instead of your finite-dimensional space of functions coming from the dot product, I will construct an infinite-dimensional space of functions which covers every way you might think of perturbing $f$.

For every point $x \in M$, pick a chart $(U_x, \varphi_x)$ with $\varphi_x(x) = 0 \in \Bbb R^n$ so that $f(U_x) \subset (f(x) - \delta, f(x) + \delta)$ for a predetermined uniform constant $\delta$, and pick a bump function $\rho$ supported in the unit ball of $\Bbb R^n$ and the identity near zero. For every $v \in T_0 \Bbb R^n$, define $$g_{x,v}(p) = \rho\left(\varphi_x(p)\right) \left(\varphi_x(p) \cdot v\right),$$ where the dot means the dot product. This function has $(dg_{x,v})_x(w) = d\varphi_x(w) \cdot v$. In particular, as $v$ varies, we see that the $(dg_{x,v})_x$ runs through the space of all functionals on $T_x M$.

---

Pick a countable set $(x_i, v_i)$ which is dense in $TM$. Let $C_n = \sum \|g_{x_i, v_i}\|_{C^n} + 2^n.$ Write $\mathcal P = \ell^1(C_n)$ for the Banach space whose elements are sequences $(a_1, \cdots)$ such that $\sum C_n |a_i| < \infty$. To each element $\pi = (a_1, \cdots)$ of $\mathcal P$ is associated a function $g_\pi: M \to \Bbb R$, given as $$g_\pi(x) = \sum a_i g_{x_i,v_i}(x).$$

The bounds on the $C^n$ norms imply that $g_\pi$ is smooth (in fact, the map $g: \mathcal P \times M \to \Bbb R$ is smooth), and further each $g_\pi$ is bounded (this comes from $C_n \geq 2^n$). So any $f + g_\pi$ is a proper smooth function.

Now consider the map $F: \mathcal P \times M \to TM$ given by $(\pi, x) \mapsto \nabla(f + g_\pi)$. You can quickly check that this map is transverse to the zero section (essentially because $\mathcal P$ is so big that it makes up all of those directions).

In particular, the "parameterized critical set" $\mathcal C \subset \mathcal P \times M$, given as $F^{-1}(0)$, is a smooth manifold; furthermore, $\mathcal C \cap \{\pi\} \times M$ is the critical set of the function $f + g_\pi$; this critical set is cut out transversely (that is,

Now apply the Sard-Smale theorem to the projection $p: \mathcal C \to \mathcal P$ to find a regular value $\pi$, and hence a smooth proper Morse function $f + g_\pi$.