Morse theory is amazing. Very geometric, more-or-less very intuitive. You don’t really explore it in detail until you’ve seen a fair bit of differential topology, but if you look closely, you start getting exposed to its core ideas as early as multivariate Calculus.
As is the fashion in modern geometry (specifically, algebraic geometry), we study geometric objects by studying the behavior of (appropriate classes of ) functions on them. But which functions? In algebraic geometry, if you’ve got some nice affine variety, you’ve got a set god-given functions to use: the coordinate ring of the variety. Here, in the affine case, this is a finitely generated, reduced (=no nilpotents) algebra over a field. Basically, a quotient of a polynomial ring by a radical ideal. Not too bad, quite manageable.
However, for a smooth manifold , the class of smooth functions on is really big. To make it “worse”, the existence of bump functions makes it hard to obtain too much cohomology info from the sheaf of smooth functions. Of course, we can use differential forms to obtain geometric (cohomological) info; this is known as the de Rham cohomology of , and it’s actually isomorphic to the singular cohomology of , which is also isomorphic to the Cech cohomology of the constant sheaf, .
Cue Morse functions.
But first (I lied), we need to recall some basic terminology. Let be a smooth function between the smooth manifolds and . For every point , induces a linear transformation between tangent spaces. We say that is a regular point of if the map is surjective (this means “is a submersion at “). If is not surjective, we say is a critical point of . Suppose . We say is a regular value of if, for all , is a regular point of . If this is not the case (i.e., some point in the preimage of is a critical point), we say that is a critical value of . If you’ve been good and remember your basic Calculus, regularity of a point/value tells us a lot (topologically) about near . Via the Implicit Function Theorem, if is a regular value, the set is a smooth submanifold of , of pure codimension one. If is a regular point of , there is an open neighborhood, of in such that is a smooth submanifold of of pure codimension one.
But what happens at critical points? Critical values? How much do we have to worry? How abundant are they? Fortunately, we have
Sard’s Theorem: the set of critical values of has measure zero in .
So, “almost all” points of are regular values of . But, let’s go deeper: what happens at critical points?
Okay, so this is where you start seeing this stuff in early Calculus. Say we’ve got a smooth function , and we look at its graph, , in . One of the first things we investigate are the “tangent lines” to points on the graph; here, these are the tangent spaces to . Using this we can answer the question “where does achieve extreme values?” Every Calc student knows (or, should know) that these can only happen (at smooth points of the domain of ) when the tangent line to at some point is “horizontal”, that is, when . Equivalently, when is not surjective (since in the one dim. case, , and is surjective iff ).
But what about the second derivative? After all, we said was infinitely differentiable. Hopefully these higher derivatives contain more information?
Of course, you already know the answer. Suppose , but . Well, it’s either going to be positive or negative. If , then we know has a local minimum at . If , then has a local maximum at . Similarly, we’d say the graph is locally “concave up” in the former case, “concave down” in the latter. Intuitively, the graph “looks like” the parabola around , depending on the sign. We can’t really apply this analysis in the case where ; for that, you’d need to use Taylor’s theorem to get more information about at .
It isn’t really until we start doing Calculus in several variables that we see the utility of this approach. Let’s move to three variables. Let be a smooth function, and let be the graph of . Suppose is a critical point of . Recall the differential in this case is given by
and saying that is a critical point of means that . Since we’ve got more than one variable, any kind of “Second derivative test” is going to need to information from all the second partial derivatives, in some way. For example, how do we reinterpret the criterion in this case?
I’ll save you the trouble and just say it: what we need to examine is something called the Hessian of f at :
The Hessian of at a point is just the matrix of second partials of , arranged in a particular way. (In the general case of , with coordinates , the Hessian takes the form . Requiring that “ now becomes , and in such a case, we say is a nondegenerate critical point of . We say
- is a local minimum of if , and ;
- is a local maximum of if , and ; and
- is a saddle point of if .
Intuitively, this says that the graph of locally looks like the paraboloid in the first two cases (depending on the sign), and like the hyperbolic paraboloid (= “saddle”) in the third case.
But what do I mean “looks like”? Is there a formal way to express this? Of course, or I wouldn’t be talking about it.
Might as well do the general case: Let be a smooth manifold of dimension , a smooth function. Let , and suppose that is a nondegenerate critical point of *. Then, there is a smooth system of coordinates about such that, in these coordinates, may be written as
where is the index of f at p (= the number of negative eigenvalues of ). This result is known as the Morse Lemma, and it legitimizes our intuition from the previous examples.
*We had previously defined the Hessian of at within a given coordinate system. As it turns out, “nondegeneracy” of a critical point is independent of coordinates, as is the index.*
Nondegeneracy of a critical point is basically the next best thing to requiring regularity of a point. In addition to the Morse lemma, nondegenerate critical points are isolated as well. That is, at such a point , we can find an open neighborhood of such that is the only critical point of . This isn’t even that hard to show: if are local coordinates about , define a new function, via
Since is a critical point of , . Then, the differential of at is equal to the Hessian of at , so nondegeneracy of implies nonsingularity of . Hence, by the Inverse Function theorem, carries some open neighborhood of in diffeomorphically onto an open neighborhood of the origin in . That is, is the only critical point of inside .
In keeping with all these definitions, we say a smooth function is a Morse function if all its critical points are nondegenerate. Some authors impose the additional requirements that every critical value has only one corresponding critical point, or that be proper (= preimage of a compact set is compact). For now, I’ll stick to my original definition.
Morse functions are basically as good as it gets for our current approach: Almost all level sets are smooth submanifolds of of codimension one, and the bad points (=critical points) where our analysis fails are isolated incidents, and even then, we know exactly what looks like in an open neighborhood of a bad point. But are Morse functions too good to be true? Do we encounter them often? As it turns out, like our worries about regular points/values, “almost all” smooth functions are Morse functions. The core of the proof is actually (again) just Sard’s theorem.
Let’s just examine the case where is a smooth function on an open subset to . Let be a choice of coordinates on . For , we define a smooth function
Theorem: No matter what the function is, for almost all choices of , is a Morse function on .
Again, we use the function . Then, the derivative of at a point is represented in these coordinates as
So, is a critical point of if and only if . Since and have the same second partials, the Hessian of at is the matrix . If is a regular value of , whenever , is nonsingular. Consequently, every critical point of is nondegenerate. Sard’s theorem then implies that is a regular value of for almost all .
There’s so much more to talk about, but I’ve already rambled on for quite a bit. Until next time.