2012-01-13

Further addendum to homework problem: As a warmup, do the following one-lower-dimensional version: Let $\Sigma$ be the embedding of $S^1$ in $\mathbb{R}^2$ shown at right and now, for each $v \in S^1$, define the analogous $f_v : S^1 \to \mathbb{R}$. Then draw a graph in $S^1 \times \mathbb{R}$ of the critical values and their indices as a function of $v$. Now the indices will only be $0$ and $1$. The critical events to note are births and deaths of pairs of critical points and crossings of critical values (one critical value rising above or below another one).

Now continuing our proof: We need a Riemannian metric on $X$, so here is the quick proof that Riemannian metrics exist.  Cover $X$ with coordinate charts $\{U_i\}$ with a corresponding partition of unity $\{ \mu_i \}$. In each coordinate chart choose the standard Euclidean inner product $g_i$. Then let $g = \sum \mu_i g_i$. This works because convex combinations of positive definite symmetric matrices are positive definite symmetric matrices.

Now note that a metric $g$, at each point $p$, is a non-degenerate bilinear form $g_p : T_p X \times T_p X \to \mathbb{R}$ and can thus equivalently be thought of as an isomorphism $g_p : T_p X \to T_p^* X$. Then using this isomorpism, we construct a vector field $W$ by $W_p = g_p^{-1}(df_p)$. Because $f$ has no critical points, $W$ is nowhere $0$. This is the gradient vector field for $f$ with respect to the metric $g$, denoted $\nabla_g f$. As a basic exercise you should verify that, when $g$ is the standard inner product on $\mathbb{R}^n$ and $f : \mathbb{R}^n \to \mathbb{R}$, then $\nabla_g f$ is the usual gradient $\nabla f$.

Now because $W$ is never $0$, $df(W) = g(W,W)$ is never $0$, so we can let $V = (1/df(W)) W$, so that $df(V) \equiv 1$. This is the vector field we wanted. Now we construct a diffeomorphism $\phi : [0,1] \times M_0 \to X$ by making $\phi(t,p)$ equal to the point $q$ you get to by starting at $p \in M_0$ and flowing forward along $V$ for time $t$. The fact that $df(V) \equiv 1$ means that $f(q) = t$, and from this and the existence and uniqueness of solutions to ordinary differential equations shows that $\phi$ is a diffeomorphism. $\Box$

So now what if there are critical points?

Suppose that $X$ is a cobordism from $M_0$ to $M_1$ with a Morse function $f : X \to [0,1]$  with one single critical point $p \in X$ of index $k$, as in the picture at right. We will again use a gradient vector field $W = \nabla_g f$ to understand the topology of $X$ in terms of the topology of $W_0$ but now, because $df_p = 0$, we cannot rescale $W$ to get a vector field $V$ with $df(V) \equiv 1$ on all of $X$. So instead we will divide $X$ into four parts, on three of which we will rescale $W$. But before we do this we need to construct our metric $g$ a little more carefully: We want there to be a coordinate chart $U$ around $p$ with respect to which $g$ is the standard Euclidean inner product and $f$ is the standard Morse local model $\sum \pm x_i^2$, so that $W = \sum \pm 2 x_i \partial_{x_i}$. This is possible because we can start with a standard Morse chart around $p$ as one of the charts in our partition of unity construction and then arrange that, in a ball neighborhood around $p$, one of the $\mu_i$'s is identically $1$ and all the others are $0$.

So now, assuming that $g$, $f$ and $W$ are standard inside a neighborhood $U$ of $p$, we draw a picture of $U$ with the level sets of $f$ and the flow lines of $V$ to the right. We choose an $\epsilon > 0$ so that $f^{-1}(f(p) - \epsilon)$ and $f^{-1}(f(p)+\epsilon)$ intersect $U$ as shown. Then our four pieces of $X$, which we will study more carefully next time, are:

1. $f^{-1}[0,f(p)-\epsilon]$, which is diffeomorphic to $[0,f(p)-\epsilon] \times M_0$ using flow along $(1/df(W))W$.
2. $f^{-1}([f(p)+\epsilon,1]$, which is diffeomorphic to $[f(p)+\epsilon,1] \times M_1$ using backward flow along $(1/df(W))W$.
3. The intersection of $f^{-1}[f(p)-\epsilon,f(p)+\epsilon]$ with the closure of the union of all flow lines for $W$ which start in some tubular neighborhood of $x_1^2+\ldots+x_k^2$ in $f^{-1}(f(p)-\epsilon)$. This is the "mystery piece" that we will understand better soon.
4. The rest of $f^{-1}[f(p)-\epsilon,f(p)+\epsilon]$, which is a product that we will also discuss next time.

2012-01-11

Addendum to the problem from last time:

In the exercise from last time, since the embedded sphere $\Sigma$ is rotationally symmetric about the $z$-axis, the critical values of $f_v$, as a function of $v \in S^2$, will be invariant under rotation of $v \in S^2$ about the $z$-axis, and thus really we might as well think of the parameter $v$ as in $S^1$, or even just in an interval from north pole to south pole in $S^2$.

So do the problem as stated, but note that your answer is rotation-invariant. Then do it again but this time using the surface $\Sigma$ at right:

Continuation of lecture:

Last lecture's definition of a Morse function was called preliminary because it did not discuss boundary behavior and compactness. Here is the full definition:

Definition: A function $f : X^n \rightarrow Y^1$ is Morse if the following conditions are satisfied:

1. For each critical point $p \in X$ there are coordinates around $p$ and $f(p)$ with respect to which $f(x_1, \ldots, x_n) = - x_1^2 - \ldots - x_k^2 + x_{k+1}^2 + \ldots x_n^2$.
2. $X$ is compact.
3. $f^{-1}(\partial Y) = \partial X$

(In class I said that either $X$ is closed and $Y$ is anything or $X$ is a cobordism (see drawing)  from $M_0$ to $M_1$ and $Y = [0,1]$ with $f^{-1}(0) = M_0$ and $f^{-1}(1) = M_1$. The way I've said it above is only slightly more general, and the cobordism case is generally the most important case to consider.)

(To say that $X$ is a cobordism from $M_0$ to $M_1$ means that $X$ is compact and $\partial X = M_0 \amalg M_1$. To say that $X$ is closed means that $X$ is compact with $\partial X = \emptyset$.)

Here are two important results about Morse functions, the proofs of which we will defer till later in the interest of getting quickly to the topological applications:

Theorem: For any compact $X$ there exists a Morse function on $X$. More precisely, for any compact $n$-manifold $X$ and any $1$-manifold $Y$, and given any function $f : \partial X \to \partial Y$, there exists an extension of $f$ to a Morse function $f : X \to Y$. (And furthermore, Morse function are generic, i.e. there are lots of them, and any given function can be perturbed in an arbitrarily small way to be Morse, more on this later.)

Thus we have Morse functions when we need them.

Theorem: If $p \in \mathbb{R}^n$ is a critical point of $f : \mathbb{R}^n \to \mathbb{R}$ such that the Hession $Hf(p)$ (the $n \times n$ matrix of second order partial derivatives) is non-degenerate as a bilinear form, then $f$ is locally Morse at $p$, i.e. there are coordinate around $p$ and $f(p)$ with respect to which $ = - x_1^2 - \ldots - x_k^2 + x_{k+1}^2 + \ldots x_n^2$. Furthermore, the index $k$ of $p$ is precisely the index of $Hf(p)$ as a bilinear form, the number of negative diagonal entries of $Hf(p)$ after diagonalizing.

Thus the index of a Morse critical point is independent of the coordinate system.

Topology from Morse functions

Our first example of recovering topological information from a Morse function is the case of a Morse function with no critical points.

Theorem: If $X$ is a cobordism from $M_0$ to $M_1$ with a Morse function $f : X \to [0,1]$, if $f$ has no critical points then $X$ is diffeomorphic to $[0,1] \times M_0$.

Proof: We will construct a vector field $V$ on $X$ such that $df(V) \equiv 1$ (which is the same thing as saying that $Df(V) = \partial/\partial y$, where I'm using $d$ for the exterial differential and $D$ for the derivative). Using this we will flow forward along $V$ from $M_0$ to construct the diffeomorphism.

To get $V$ we need a Riemannian metric (there are other more direct ways using a partition of unity to directly patch together such $V$'s on coordinate charts, but using a Riemannian metric has some advantages and is at the very least an important idea). A Riemannian metric $g$ on $X$ is a choice of an inner product $g_p$ on $T_p X$ for each $p \in X$, varying smoothly in $p$. (Varying smoothly in $p$ just means that, when $g$ is written as an $n \times n$ matrix in local coordinates, the entries of the matrix are all smooth functions of $p$.) Next time we will use a partition of unity to show that Riemannian metrics exist and then show how to use such a metric to get $V$.