Stability and Genericity

Before I begin, I want to give credit where credit is due: much of the exposition (especially the proofs) of my last post was paraphrased from Guillemin and Pollack’s Differential Topology [1]. One of my favorites.

Okay, moving on.

We saw last time that Morse functions are pretty neat, and are abundant; “almost all” smooth functions are actually Morse functions. I’d like to take a minute to talk about this type of property (along with a related notion, “genericity“), as well as a notion called stability. “Almost all” is usually a phrase one comes across in analysis (or, as we saw, fields that use analysis, of which there are tons), and it means “the set of ‘bad choices’ is a set of (some suitable) measure zero.” “Genericity,” or, “being generic” is more or less the algebro-geometric counterpart to “almost all” (although it isn’t uncommon to use “generic” to mean “almost all”). Something is generic in something else if it is true on an open dense set. In algebraic geometry, we’d usually say “the set of bad choices lies in a subset of strictly smaller dimension.” Anyway, the basic idea is that such properties/objects are what you’d expect to find if you picked one “at random.” For example: draw a curve on a piece of paper (and pretend it’s $\mathbb{R}^2$ ). If you were to close your eyes, and put your finger on the paper, you’d basically always miss the curve, and land on blank space, illustrating that a generic point of $\mathbb{R}^2$ isn’t on the curve.

What do I mean by stability? What is “stable?” If you recall the sketch of a proof I gave last time for “almost all functions are Morse functions,” given some smooth functions $f: U \to \mathbb{R}$ (where $U \subseteq \mathbb{R}^n$ is an open subset), we “deformed,” or “perturbed” $f$ into a Morse function, $f_a := f + a_1 x_1 + \cdots +a_n x_n$ , by adding a generic linear form. If I deform some smooth map $f_0: M \to N$ to another map $f_1 : M \to N$ , I’m invoking one the fundamental operations in (differential) topology: homotopy. We’d say $f_0$ and $f_1$ are homotopic, usually written $f_0 \thicksim f_1$ , if there exists a smooth map $F: M \times [0,1] \to N$ such that, for all $x \in M$ , $F(x,0) = f_0(x)$ and $F(x,1) = f_1(x)$ . Smoothness of $F$ then ensures that all the “in between” maps $f_t(x) := F(x,t)$ are smooth as well. Here’s a simple example to illustrate that this is really what we mean when we say $f_0$ is deformed to $f_1$ .

Here’s a really, really nice description of this notion in [1]:

In the real world of sense perceptions and physical measurements, no continuous quantity or functional relationship is ever perfectly determined. The only physically meaningful properties of a mapping, consequently, are those that remain valid when the map is slightly deformed. Such properties are stable properties, and the collection of maps that posses a particular stable property may be referred to as a stable class of maps. Specifically, a property is stable provided that whenever $f_0: X \to Y$ possesses the property and $f_t : X \to Y$ is a homotopy of $f_0$ , then, for some $\epsilon > 0$ , each $f_\epsilon$ with $t < \epsilon$ also possesses the property.

In this vein, the idea is that stable properties are “observable.” These are the types of things we want to look for when playing around with functions.

I said before that the ideas of stability and genericity were related. Suppose I want to find a Morse function $f: M \to \mathbb{R}$ . We know already that almost all smooth, real valued functions on $M$ are Morse. But what happens if I happen to pick a bad one? Never fear; we deform $f$ by adding some generic linear form $\ell_a = \sum_{i=1}^n a_i x_i$ . Moreover, by the genericity of the choice of $\ell_a$ , we can pick good choices of “deformation vector” $a = (a_1,\cdots,a_n)$ such that $\| a \|$ is arbitrarily small. Hence, even if we end up picking a bad function $f$ , for any $\epsilon > 0$ , we can find a Morse function $f_a$ such that $\|f - f_a\| < \epsilon$ . This is a common occurrence for stable properties: even if you happen to find a bad function, there are arbitrarily close good functions (in the space of smooth maps with, say, the supremum norm). Some of the most common stable properties for a smooth map $f: M \to N$ are:

local diffeomorphisms
immersions
submersions
maps transversal to a given submanifold $Q \subseteq N$
embeddings
diffeomorphisms.

Morse functions are also stable, with the caveat that we require our domain to be compact. Let $f$ is a Morse function on a compact manifold $X$ , and let $f_t$ be a homotopic family of functions with $f_0 = f$ . Then, $f_t$ is Morse for all $t$ sufficiently small.

In upcoming events, we’ll want to analyze the topology of a manifold by studying the level sets of Morse functions on the them, and these notions of genericity and stability will ensure that the selection of such functions is never in short supply.

Stability and Genericity

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