Morse and Cerf Theory
20120109
Morse Theory studies the topology of smooth manifolds by looking at generic smooth maps (Morse functions) from smooth manifolds to the reals (or, sometimes, to the circle), and investigating the critical points of such maps and their indices, interactions, the gradient flow lines for the maps, etc. Cerf Theory studies smooth 1parameter families of functions connecting different Morse functions on the same smooth manifold. Although there are important infinitedimensional versions of these theories, I will focus on the finitedimensional setting and, especially, on low dimensions like 2, 3, 4 and 5. I will focus less on the foundational analytic technicalities and more on the applications. Some of the most important results I want to get to are (not necessarily in this order, and not necessarily in full detail):

The classification of surfaces.

The existence of Heegaard splittings of 3manifolds.

Generators and relations for the mapping class groups of surfaces.

Handlebody decompositions for 4manifolds and surgery diagrams for 3manifolds.

The Kirby calculus for 3manifolds and 4manifolds.

The isotopy versus pseudoisotopy problem as studied by Cerf and HatcherWagoner.

Understanding Morse 2functions (generic maps to 2manifolds).
We begin with a problem: Consider the embedding of $S^2$ in $\mathbb{R}^3$ shown at left. This is supposed to be rotationally symmetric about the red $z$axis. Let $\Sigma$ denote this particular submanifold of $\mathbb{R}^3$ (diffeomorphic to $S^2$). Now let $f : \Sigma \rightarrow \mathbb{R}$ be orthogonal projection onto the $z$axis. Note that $f$ has two critical points, indicated, a maximum and a minimum. Now note that, for any unit vector $v \in S^2$, we can project $\Sigma$ orthogonally onto the oriented line spanned by $v$ to get another function $f_v : \Sigma \rightarrow \mathbb{R}$. Thus our original $f$ is $f_{(0,0,1)}$, and $f$ is $f_{(0,0,1)}$. (We are assuming the origin is in the middle of the picture at left.) For another example, $f_{(1,0,0)}$ will have six critical points: two maxima, two minima and two saddles, as in the figure below:
Thus we have a 2parameter family of functions $f_v$, parameterized by the 2dimensional parameter space $S^2$. The problem is to somehow depict with a diagram the behavior of the critical values of $f_v$ and their indices as $v$ ranges over $S^2$. (The index of a minimum is $0$, the index of a saddle is $1$, the index of a maximum is $2$, more on indices later.)
(Recall: for a smooth $f : X \rightarrow Y$, a point $p \in X$ is a critical point if $Df_p : T_p X \rightarrow T_{f(p)} Y$ does not have maximal rank. If $p \in X$ is a critical point, then $f(p) \in Y$ is a critical value. When $Y$ is 1dimensional, not having maximal rank simply means being equal to 0.)
One possible way to depict this is to draw a diagram in $S^2 \times \mathbb{R}$, where in each $v \times \mathbb{R}$ you draw the critical values of $f_v$, labelled with their indices. Since the critical values are usually isolated, you should have some kind of surface in $S^2 \times \mathbb{R}$, possibly with some interesting singularities. A convenient picture for $S^2 \times \mathbb{R}$ is an open shell between two concentric spheres (identifying $\mathbb{R}$ with an open interval).
This picture drawn in $S^2 \times \mathbb{R}$ can also be thought of as the set of critical values of the function $F : S^2 \times \Sigma \rightarrow S^2 \times \mathbb{R}$ given by $(v,p) \mapsto (v,f_v(p))$.
We are just about to define a Morse function properly, given the above preamble, but first we mention one famous problem that was studied using Morse and Cerf theory:
An isotopy between two maps $f_0, f_1 : X \rightarrow Y$ can be define as a diffeomorphism $F : [0,1] \times X \rightarrow [0,1] \times Y$ such that $F(0,p) = (0,f_0(p))$, $F(1,p) = (1,f_1(p))$ and $F$ is "levelpreserving", i.e. $F$ is the identity on the $[0,1]$ component, or $F(t,p) = (t,f_t(p))$. A pseudoisotopy between $f_0$ and $f_1$ is a diffeomorphism $F : [0,1] \times X \rightarrow [0,1] \times Y$ satisfying the first two criteria but not necessarily levelpreserving. Note than when gluing manifolds together along boundaries, the resulting manifold is determined up to diffeomorphism by the pseudoisotopy class of the gluing map (this is a good exercise to prove), but in other contexts the difference between isotopy and pseudoisotopy is very important. Cerf studied 1parameter families of functions connecting Morse functions in order to understand this problem, and hopefully we'll get to that later.
Now:
Preliminary definition: A Morse function is a smooth map $f$ from an $n$manifold $X$ to a $1$manifold $Y$ such that, for every critical point $p \in X$, there exist local coordinates $(x_1, \ldots, x_n)$ about $p$ and a coordinate $y$ about $f(p)$ with respect to which $f(x_1,\ldots, x_n) = x_1^2  \ldots  x_k^2 + x_{k+1}^2 + \ldots x_n^2$. The integer $k$ is called the index of $p$.
We need to see that Morse functions exist, that the index is independent of the coordinates, that there are lots of Morse functions, that the property of being Morse is stable (doesn't change under small perturbations), and many other foundational facts. But first some examples:
When $n=2$, we have minima $f(x_1,x_2) = x_1^2 + x_2^2$ with index $k=0$, saddles $f(x_1,x_2) = x_1^2 + x_2^2$ with index $k=1$ and maxima $f(x_1,x_2) = x_1^2  x_2^2$ with index $k=2$.
When $n=1$ we have minima $f(x) = x^2$ with index $k=0$ and maxima $f(x)=x^2$ with index $k=1$.
Note that $f(x) = x^3$ is not Morse, and that there is an interesting perturbation $f_t(x) = x^3 + tx$. When $t<0$, $f_t$ is Morse with one min and one max, and when $t>0$, $f_t$ is Morse with no critical points. We'll discuss these phenomena more carefully soon, and they do arise in the problem that we began with.