# Tuesday February 4th

## Modifying Vector Fields

Recall:
Let $f: W \to I$ be Morse, $\crit(F) = \theset{p, q}$ where $f(p) < f(q)$, and $\xi$ a gradient-like vector field for $f$.

Theorem (When Critical Points of Morse Functions Can Take On Any Value)
: If $W^*(p) \intersect W^*(q) = \emptyset$ then for any $a, b\in (0 ,1)$ we can change $f$ "nicely" to a new Morse function $g$ such that $g(p) = a$ and $g(q) = b$.

![Image](figures/2020-02-04-11:09.png)\

Note that these are disjoint iff $W^u(p) \intersect W^s(q) = \emptyset$ iff $S_R^c(p) \intersect S_L^c(q) = \emptyset$.
If $\in(p) \geq \in (q)$ then
$$
\dim S_R^c(p) = \dim S_L^c(q) < n-1 = \dim f\inv(c)
.$$

![Image](figures/2020-02-04-11:13.png)\

Lemma (1, Small Submanifolds Are Disjoint Up to Isotopy)
: For $M^m, N^n \subset V^v$ submanifolds with $m+n < v$, there exists a diffeomorphism $h: V \to V$ smoothly isotopic to $\id_V$ such that $h(M) \intersect N = \emptyset$.

> I.e. low enough dimension submanifolds can smoothly be made disjoint.

Lemma (2, ??)
:   Let $f: W\to I$ be Morse with gradient-like vector field $\xi$ and regular value $x\in (0, 1)$.
    Let $h: f\inv (c) \to f\inv(c)$ be smoothly isotopic to the identity, and define $M \definedas f\inv(c)$.
    Then we can change $\xi$ over $f\inv [c-\eps, c]\lambda$ to a new gradient-like vector field $\bar \xi$ such that if we let
    $$
    \Phi: f\inv(c-\eps) \to M
    $$
    be the flow induced by $\xi$ and
    $$
    \bar\Phi: f\inv(c-\eps) \to M
    $$
    be induced by $\bar \xi$.

    ![Image](figures/2020-02-04-11:22.png)\

    > Note that the left/right spheres are defined in terms of gradient-like vector fields, so "bar" here refers to a new gradient-like vector field.

    Then picking $h$ from lemma 1, we can arrange so that $\bar S_L^c(q) \intersect \bar S_R^c(p) = \emptyset$.

    Then $\bar \xi = h\circ \Phi$.

### Proof of Lemma 2

We have
$$
[c-\eps, c] \cross M \mapsvia{\phi} f\inv[c-\eps, c] \mapsvia{f} [c-\eps, c]
,$$
which we can factor by projection onto the first component.
This satisfies the following properties:

1. $\phi_*(\dd{}{t}) = \hat \xi \definedas \frac{1}{\xi(f)} \xi$
2. $\restrictionof{\phi}{\theset{x} \cross M} = \id$

> Note: the product cobordism $[c-\eps, c] \cross M$ is easier to work with here, we can then push it to $f\inv[c-\eps, c]$ via $\phi$.

We now define $h_t$ by the properties

- For $t$ near $c-\eps$, $h_t = \id$, and
- For $t$ near $x$, $h_t = h$.

We use this to construct a diffeomorphism

\begin{align*}
H: [c-\eps, c] \cross M &\to [c-\eps, c] \cross M \\
(t,x) &\mapsto (t, h_t(x))
.\end{align*}

Both the domain and codomain map via $\phi$ to $f\inv[c-\eps, c]$, so we can consider

\begin{align*}
H_*\qty{ \dd{}{t}}
= \dd{H}{t} (t, x) = \qty{ 1, \dd{h_t}{t}(x)} \quad\text{for $t$ near } c-\eps \text{ and } c
.\end{align*}

We then have $\xi' = (\phi \circ H \circ \phi\inv)_* \hat \xi = (\phi\circ H)_* \qty{\dd{}{t}}$.
Thus for $t$ near $c-\eps$ and $c$, we have $\xi' = \phi_*\qty{\dd{}{t}} = \hat \xi$.
So define

\begin{align*}
\bar \xi =
\begin{cases}
\xi(f) \cdot \xi' & \text{ on } f\inv[c-\eps, c] \\
\xi & \text{ everywhere else }
\end{cases}
.\end{align*}

On $[c-\eps, c] \cross M$, what are the integral curves of $H_*(\dd{}{t})$?
Picking a $t\in [c-\eps, c]$, we have $H(t,x) = (t, h_t(x))$ by definition, and thus the integral curves of $\hat \xi$ are given by $\phi(t, h_t(x))$ for all $x\in M$.
Then $\phi(c-\eps, x) = \phi(c-\eps, h_{c-\eps}(x))$ for $t=c-\eps$, which is just $\Phi\inv(x)$.
Then for $t=c$ we get $\phi(c, h(x)) = h(x)$.
Thus $\bar \Phi \circ \Phi\inv (x) = h(x)$, yielding $\bar \Phi = h\circ \Phi$.

Corollary
:   Given any Morse function $f$ on an $n\dash$dimensional cobordism $(W^n; M_0, M_1)$ we can get a new Morse function $g$ such that

    - $\crit(g) = \crit(f)$,
    - $g(p) = \ind(p)$ (since we can make the critical points take on any values)
    - $g\inv(-1/2) = M_0$ and $g\inv(n+1/2) = M_1$


    Such a Morse function is called *self-indexing*.

## Cancellation

Note that we may have "extraneous" critical points:

![Image](figures/2020-02-04-11:52.png)\

Here note that $W$ is diffeomorphic to a product cobordism.

Let $f: W \to [0, 1]$ be Morse, $\crit(f) = \theset{p, q}$, $\ind(p) = \lambda$ and $\in(q) = \lambda + 1$ with $f(p) < f(q)$.

Pick $c\in [f(p), f(q)]$, then consider $S_R^c(p) \intersect S_L^c(q)$.
We have $\dim S_R^c(p) = n-\lambda - 1$ and $\dim S_L^c(q) = \lambda + 1 - 1$, and so the dimension of their intersection is $n-1$, i.e. $\dim f\inv(c)$.

Definition (Transverse Submanifolds)
: Submanifolds $M^m, N^n \subset V^v$ are called *transverse* if for any $p\in M\intersect N$, $T_pV \subset \spanof\theset{T_p M, T_p N}$ and we write $M \transverse N$.

Example
:   - If $m+n < v$, then $M\transverse N$ iff $M\intersect N = \emptyset$.
    - If $m+n = v$, then $M\transverse N$ iff $\dim M\intersect N = 0$.
    - In general, if $M\transverse N$ then $M\intersect N$ is a smooth submanifold of dimension $m+n -v$.

Theorem (Submanifolds are Transverse up to Isotopy)
: For submanifolds $M_m, N^n \subset V$, then there exist $h: V \mapsvia{\text{diff}} V$ smoothly isotopic to $\id_V$ such that $h(M) \transverse N$.

Corollary
:   We can change $\xi$ in $f\inv[c-\eps, c]$ such that $S_R^c(p) \transverse S_L^c(q)$, so their intersection consists of finitely many points.

    ![Image](figures/2020-02-04-12:08.png)\

Proposition (First Cancellation)
: If $S_R^c(p)$intersects $S_L^c(q)$ in exactly one point, then $W$ is a product cobordism.


Idea of proof:

1. Change $\xi$ in a neighborhood of the integral curve from $p$ to $q$ such that the new vector field is nonvanishing.
2. Change $f$ to $g$ with no critical point such that the new vector field is gradient-like for $\xi$.