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In the strong $C^{\infty}$ topology on $C^{\infty} (M, R)$ , the Morse functions form a residual subset

Statement

Let $M$ be a smooth manifold. In the strong $C^{\infty}$ topology on $C^{\infty} (M, R)$ , the set of Morse functions is residual.

Facts & Assumptions

Given: A smooth manifold $M$ .

[F1]

A smooth 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).

[L1]

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

[L2]

Every strong neighbourhood of a smooth function contains a Morse perturbation, and that perturbation can be chosen to agree with the original function near any prescribed closed region where transversality already holds (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).

[A1]

For a fixed compact set $K$ , the condition that a differential section be transverse to the zero section on a neighbourhood of $K$ is open in the strong topology, because only finitely many first derivatives on a compact neighbourhood of $K$ are involved.

Proof

technique · direct

1.1

By [L1], choose a smooth proper exhaustion $h : M \to [0, \infty)$ and write $K_{n} : = h^{- 1} ([0, n])$ . For each $n \geq 1$ , let $O_{n}$ be the set of smooth functions $g$ such that $d g$ is transverse to the zero section on some open neighbourhood of $K_{n}$ . If $g$ is Morse, then [F1] gives $g \in O_{n}$ for every $n$ . Conversely, if $g \in ⋂_{n} O_{n}$ , then every point of $M$ lies in some $K_{n}$ , so $d g$ is transverse near that point; [F1] then makes $g$ Morse. Thus the Morse functions are exactly $⋂_{n \geq 1} O_{n}$ .

F1L1givenconstruct

2.1

Each $O_{n}$ is open by [A1].

A1step 1.1

2.2

Each $O_{n}$ is dense. Indeed, given any smooth function $f$ and any strong neighbourhood $U$ of $f$ , [L2] produces a Morse function $g \in U$ , and step 1.1 shows that every Morse function belongs to every $O_{n}$ .

L2step 1.1

3.1

Step 1.1 identifies the Morse functions with a countable intersection of the open dense sets $O_{n}$ from steps 2.1 and 2.2. Therefore the Morse functions form a residual subset of $C^{\infty} (M, R)$ in the strong topology.

step 1.1step 2.1step 2.2∎

Depends on

Used by

On a noncompact manifold, this page states Morse genericity as a strong-topology residual theorem Remark

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, Part 10 (standard reference, not scraped)
Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)

In the strong C∞C∞ topology on C∞(M,R)C∞(M,R), the Morse functions form a residual subset