# Thursday January 16th

## Approximation with Morse Functions with Distinct Critical Points

Theorem (Morse Functions and Distinct Critical Points)
:   Let $f: M \to \RR$ be Morse with critical points $p_1, \cdots, p_k$.
    Then $f$ can be approximated by a Morse function $g$ such that

    1. $g$ has the same critical points of $f$
    2. $g(p_i) \neq g(p_j)$ for all $i\neq j$.


Idea: Change $f$ gradually near critical points without actually changing the critical points themselves.

## Proof of Theorem

Suppose $f(p_1) = f(p_2)$.

Choose $\bar U \subset N$ open neighborhoods of $p_1$ such that $\bar N$ doesn't contain $p_i$ for any $i$ except for 1.
Note that this is possible because the critical points are isolated.

![Image](figures/2020-01-16-11:05.png)\

Choose a bump function $\lambda \equiv 1$ on $U$ and $0$ on $M\setminus N$.
Now let $f_1 = f + \varepsilon \lambda$, where we'll see how to choose $\varepsilon$ small enough soon.

Let $K \definedas \theset{ x \suchthat 0 < \lambda(x) < 1 }$, which is compact.

Pick a Riemannian metric on $M$, then we can talk about gradients.
Recall that $\grad f$ is the vector field that satisfies $\inner{X}{f}$ for all vector fields $X$ on $M$.
Because $f$ has no critical points in $K$, $X(f)$ is nonzero for some field $X$, so $\grad f$ is nonzero, noting that $\grad f$ is only zero at the critical points of $f$.

In particular, on $K$ we have $0 < c \leq \abs{\grad f}$ for some $c$, and $\grad \lambda \leq c'$ for some $c'$.
So pick $0 < \varepsilon  < c'/c$ such that $f_1(p_1) \neq f_1(p_2)$, $f_1(p_1) = f(p_1) + \varepsilon$, and$f_1(p_i) = f(p_i)$ for all $i\neq 1$.
Note that this is possible because there are only finitely many points, so almost every $\varepsilon$ will work.

Claim 1
:   The critical points of $f_1$ are exactly the critical points of $f$.

Proof
:   In $K$, we have
    \begin{align*}
    \grad f_1 = \grad f + \varepsilon \grad \lambda \implies \abs{\grad f_1} \geq \abs{\grad f} - \varepsilon \abs{\grad \lambda} \geq x - \varepsilon c' > 0
    .\end{align*}

    If $x\not\in K$, we have

    1. $x\in U$, or
    2. $x\in M\setminus N$

    In case 1, $\lambda$ is constant and $\grad \lambda = 0$, so $\grad f_1 = \grad f$.
    In case 2, $\lambda$ is again constant, so the same conclusion holds.

Claim 2
: $f_1$ is Morse.

Proof
:   In a neighborhood of $p_1$, we have $f_1 \equiv f + \varepsilon$.
    In a neighborhood of $p_i$, we have $f_1 \equiv f$.

    We can then check that $J_{f_1}(p_i) = J_f(p_i)$, and since $f$ is Morse, $f_1$ is Morse as well.

Recall that this lets us put an order on $f(p_i)$.
Between every critical value, pick regular values $c_i$, i.e. $f(p_1) < c_1 < f(p_2) < \cdots$.
Then $f\inv(c_i)$ is a smooth submanifold of dimension $n-1$, and we have the following schematic:

![Image](figures/2020-01-16-11:26.png)\

Moreover, $f\inv[c_i, c_{i+1}]$ is a cobordism from $f\inv(c_2)$ to $f\inv(c_{i+1})$.

Definition (Morse Functions for Cobordisms)
:   Recall that for $(W; M_0, M_1)$ a cobordism, a Morse function $f: W \to [a, b]$ is Morse iff

    1. $f\inv(a) = M_0$ and $f\inv(b) = M_1$.
    2. $f$ has only nondegenerate critical points and no critical points near $\bd W = M_1 \disjoint M_2$, i.e. all critical points are in $W^\circ$ (the interior).

Proof of density of Morse functions goes through in the same way, with extra care taken to choose neighborhoods that do not intersect $\bd W$.

Theorem (Cobordisms, Morse Functions, Distinct Critical Points)
:   1. For every cobordism $(W; M_1, M_2)$ there exists a Morse function.
    2. The set of such Morse functions is dense in the $C^2$ topology.
    3. Any Morse function $f: (W; M_1, M_2) \to [a, b]$ can be approximated by another Morse function $g: (W; M_1, M_2) \to [a, b]$ such that $g$ has the same critical points of $f$ and $g(p_i) \neq g(p_j)$ for $i\neq j$ (i.e. the critical points are distinct).

Note that $n\dash$manifolds are a special cases of cobordisms, namely a manifold $M$ is a cobordism $(W; M, \emptyset)$.
So all statements about cobordisms will hold for $n\dash$manifolds.

Definition (Morse Number)
: The **Morse number** $\mu$ of a cobordism $(W; M_0, M_1)$ is the minimum of $\abs{\theset{ \text{critical points of } f \suchthat f \text{ is Morse } }}$.

We'll be considering cobordisms with $\mu = 0$.

> Note: if we take $X = \grad f$, we have $\inner{X}{\grad f} = \norm{\grad f}^2 \geq 0$, which motivates our next definition.

Definition (Gradient-Like Vector Fields)
:   Let $f: W \to [a, b]$ be a Morse function.
    Then a **gradient-like vector field** for $f$ is a vector field $\xi$ on $W$ such that

    1. $\xi(f) > 0$ on $W\setminus\crit(f)$.
    2. For every critical point $p$ there exist coordinates $(x_1, \cdots, x_n)$ on $U \ni p$ such that

    \begin{align*}
    f(X) = f(p) - x_1^2 - \cdots - x_{\lambda^2} + x_{\lambda + 1}^2 + \cdots + x_n^2
    ,\end{align*}

    as in the Morse Lemma, where $\lambda$ is the index, and

    \begin{align*}
    \xi = (-x_1, -x_2, \cdots, -x_\lambda, x_{\lambda+1}, \cdots, x_n) \text{ in } U
    .\end{align*}

Lemma (Morse Functions Have Gradient-Like Vector Fields)
: Every Morse function $f$ on $(W; M_0, M_1)$ has a gradient-like vector field.

## Proof: Every Morse Function has a Gradient-Like Vector Field

For simplicity, assume $f$ has a single critical point $p$.
Pick coordinate $(x_1, \cdots, x_n)$ on an open set $U_0$ around $p$ such that $f$ has the form given in (1) above.
Define $\xi_0$ on $U_0$ to be (2) above.

Every point $q\in W\setminus U_0$ has a neighborhood $U'$ such that $df\neq 0$ on $U'$.
By the implicit function theorem, there is a smaller neighborhood $U''$ such that $q \in U'' \subset U$ such that $f= c_0 + x_1$ on $U''$ for some constant $c_0$.

> Exercise: check that this works!

But since $W\setminus U_0$ is a closed subset of a compact manifold, it is compact, so we can cover it with finitely many $U_i$ that satisfy

1. $U_i \intersect U = \emptyset$ for some open $U$ containing $p$ such that $U\subset U_0$ and $\bar U \subset U_0$.
2. $U_i$ has a coordinate chart $(x_1^2, \cdots, x_n^2)$ such that $f = c_i + x_1^2$ on $U_i$ for some constants $c_i$.

Thus on $U_i$ we can set $\xi_i = (1, 0, \cdots, 0) = \dd{}{x_1^2}$ to get local vector fields.
We can then take a partition of unity $\rho_1, \cdots, \rho_k$ and set $\xi = \sum_i \rho_i \xi_i$.

Now consider $\xi(f)$.
By definition, $\xi(f) = \sum_i \rho_i \xi_i (f)$.
Note that $\rho_i \xi_i (f) = 1$ in $U_i$, and $\rho_0 \xi_0 (f) \geq 0$, so $\xi(f) \geq 0$.
If $x$ is not a critical point, then at least 1 $\xi_i(f)(x)$ is positive and thus $\xi(f)(x) > 0$.

> This is because $x$ is either in $U$, in which case the 0 term is positive, or $x \in U_i$, in which case one of the remaining terms is positive.

$\qed$

> The idea here: if we can make locally gradient-like vector fields, we can use partitions of unity to extend them to global vector fields.

Theorem (The Morse Number Detects Product Cobordisms)
:   Any cobordism $(W; M_0, M_1)$ with $\mu = 0$ is a product cobordism, i.e.

    \begin{align*}
    (W; M_0, M_1) \cong ( M_0 \cross I; M_0 \cross \theset{0}, M_0 \cross \theset{1} )
    .\end{align*}

Proof (of Theorem)
:   Let $f: W \to I$ be Morse with no critical points, and let $\xi$ be a gradient-like vector field for $f$.

    Then $\xi(f) > 0$ on $W$, so we can normalize to replace $\xi$ with $\frac{1}{\xi(f)} \xi$ and assume $\xi(f) = 1$.
    Then consider the integral curves of $\xi$, given by $\phi:[a, b] \to W$.

    > i.e. $d\phi = \xi$.

    We can thus compute
    \begin{align*}
    \dd{}{t} f\circ \phi(t) = df(\dd{\phi}{t}) = df(\xi) = \xi(f) = 1
    .\end{align*}
    By the FTC, this implies that $f\circ \phi(t) = c_0 + t$ for some constant $c_0$.
    So reparameterize by defining $\psi(s) = \phi(s - c_0)$, then $f\circ \psi(s) = s$.
    For every $x\in W$, there exists a unique maximal integral curve $\psi_x(s)$ that passes through $x$.

    > Note that this works because maximal curves must intersect the boundary at precisely $t=0, 1$ and $f$ is an increasing function.
    > So for any curve passing through $x$, we can extend it to a maximal.

    ![Image](figures/2020-01-16-12:21.png)\

    We can then define

    \begin{align*}
    h: M_0 \cross I &\to W \\
    (x, s) &\mapsto \psi_x(s) \\
    (\psi_y(0), f(y)) &\mapsfrom y\\
    .\end{align*}