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Every smooth manifold admits a proper Morse function

Statement

Every smooth manifold admits a proper Morse function.

Facts & Assumptions

Given: A smooth manifold $M$ .

[L1]

Every smooth manifold admits a smooth proper exhaustion function (Every smooth manifold admits a smooth proper exhaustion function).

[L2]

One can perturb a smooth function to a Morse function in an arbitrarily small strong neighbourhood, while fixing any closed region where the differential is already transverse to the zero section (Every smooth function admits arbitrarily fine strong-topology perturbations whose differential is transverse to zero, supported away from a closed set where transversality already holds).

[F1]

A function is Morse exactly when its differential section is transverse to the zero section (A smooth function is Morse if and only if its differential section is transverse to the zero section).

[A1]

The set of smooth functions $g$ satisfying $∣ g (x) - h (x) ∣ < \frac{1}{4}$ for every $x \in M$ is a strong $C^{\infty}$ neighbourhood of $h$ . Equivalently, proper maps form an open subset in the strong topology, as recorded in the cited Frejlich notes.

Proof

technique · direct

1.1

Choose a smooth proper exhaustion $h : M \to [0, \infty)$ by [L1], and let $U : = {g \in C^{\infty} (M, R) : ∣ g (x) - h (x) ∣ < \frac{1}{4} for every x \in M} .$ By [A1], this is a strong neighbourhood of $h$ .

L1A1givenchoose

2.1

Apply [L2] to the function $h$ , the closed set $A : = \emptyset$ , and the strong neighbourhood $U$ . It yields a smooth function $g \in U$ whose differential is transverse to the zero section.

L2step 1.1

3.1

Since $g \in U$ , one has $g \geq h - \frac{1}{4}$ . Thus, for every real $c$ , ${x : g (x) \leq c} \subseteq {x : h (x) \leq c + \frac{1}{4}} .$ The left-hand side is closed and the right-hand side is compact because $h$ is proper, so every sublevel set of $g$ is compact. Also $g \geq - \frac{1}{4}$ , hence the inverse image under $g$ of every compact subset of $R$ is a closed subset of one of these compact sublevel sets. Therefore $g$ is proper. Since step 2.1 gives $d g$ transverse to the zero section everywhere, [F1] makes $g$ Morse.

F1step 1.1step 2.1algebra

4.1

Hence $g$ is a proper Morse function on $M$ .

step 3.1∎

Depends on

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Sources

Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)
Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)
Pedro Frejlich, Morse Theory, Lecture Two (standard reference, not scraped)