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:T2EmbbededInR3

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) CobordismMorseFcn 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. ProductFlow

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 $V$.