Over the past few decades, Morse theory has undergone many generalizations, into many different fields.  At the moment, I only know of a few, and I understand even fewer. Well, let’s begin at the beginning:

  • Classical Morse theory (CMT)
  • Stratified Morse theory (SMT)
  • Micro-local Morse theory (MMT)

The core of these theories is, of course, the study of Morse functions on suitable spaces and generalizations/interpretations of theorems in CMT to these spaces.  For CMT, the spaces are smooth manifolds (or, compact manifolds, if your definition of Morse function doesn’t require properness).  SMT looks at Morse functions on (Whitney) stratified spaces, usually real/complex varieties (either algebraic or analytic), and more generally, subanalytic subsets of smooth manifolds.  MMT deals with both cases, but from a more “meta” perspective that I’m not going to tell you about right now.

The overarching theme is pretty simple:  one can investigate the (co)homology of X by examining the behavior of level sets of Morse functions as they “pass through” critical values.  First, we’ll need some notation.  Let M be a smooth manifold, a < b \in \mathbb{R}, and let f: M \to \mathbb{R} be a smooth function.  Then, set

  • M_{\leq a} := f^{-1}(-\infty,a]
  • M_{< a} := f^{-1}(-\infty,a)
  • M_{[a,b]} := f^{-1}[a,b]

In CMT, this overarching idea is described by two “fundamental” theorems:

Fundamental Theorem of Classical Morse theory, A (CMT;A):

Suppose f has no critical values on the interval [a,b] \subseteq \mathbb{R}.  Then, M_{\leq a} is diffeomorphic to M_{\leq b}, and the inclusion M_{\leq a} \hookrightarrow M_{\leq b} is a homotopy equivalence (that is, M_{\leq a} is a deformation-retract of M_{\leq b}).

Homologically speaking, this last point can be rephrased as H_*(M_{\leq b},M_{\leq a}) = 0 (for singular homology with \mathbb{Z} coefficients).

Fundamental Theorem of Classical Morse theory, B (CMT;B):

Suppose that f has a unique critical value v in the interior of the interval [a,b] \subseteq \mathbb{R}, corresponding to the isolated critical point p \in M of index \lambda.  Then, H_k(M_{\leq b},M_{\leq a}) is non-zero only in degree k = \lambda, in which case $latex H_\lambda(M_{\leq b},M_{\leq a}) \cong \mathbb{Z}$.

So, if c \in \mathbb{R} varies across a critical value a < v < b of f, the topological type of M_{\leq c} “jumps” somehow.  If we want to compare how topological type of M_{\leq b} differs from that of M_{\leq a}, the obvious thing to do is consider them together as a pair of spaces (M_{\leq b}, M_{\leq a}) and look at the relative (co)homology of this pair.  CMT;A and CMT;B together tell us that we’re only going to get non-zero relative homology of this pair when there is a critical value between a and b, and in that case, the homology is non-zero only in degree \lambda.

But HOW does the topological type change, specifically, as we cross the critical value?

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.

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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.