# Tuesday February 18th

## Cancellation Theorems

Theorem (Rearrangement)
: If a Morse function $f$ has 2 critical points with $f(p) < c < f(q)$ and $\ind(p) \geq \ind(q)$, then $\xi$ can be perturbed in a neighborhood of $f\inv(c)$ such that $W_p^s \intersect W_q^u = \emptyset$.

Theorem (First Cancellation)
: If $S_R \transverse S_L' = \theset{\pt}$, then the cobordism is diffeomorphic to a product.

Theorem (Second Cancellation)
: Suppose $(W, V_0, V_1)$ is a cobordism and $f: W\to \RR$ has two critical points.
If $S_R \cdot S_L' = \pm 1$, then $W^n \cong V_0 \cross [0, 1]$.

Theorem (Whitney's Trick)
:   If $M, M' = M^m, M^n \subset V^{m+n}$ are closed submanifolds with $M\transverse M'$ such that $M, \nu(M')$ (the normal bundle) are oriented.
    Assume $m+n \geq 5$ and $n\geq 3$, and if $m=1,2$ then assume $\pi_1(V\setminus M') \injects \pi_1(V)$.
    Let $p, q$ be in the intersection, $\eps(p)\cdot \eps(q) = -1$ (local intersection numbers), and there exists a contractible loop $L\subset V$ such that

    - $L = L_0 \union L_1$
    - $L_0$ is smooth in $M$, $L_1$ is smooth in $M'$.
    - $L_0, L_1$ go from $p$ to $q$.

    And suppose each $L_i \intersect (M\intersect M' \setminus{p, q}) = \emptyset$.

    Then

    1. $h_0 = \id$,
    2. $h_+ = \id$ near $M \intersect M' \setminus \theset{p, q}$
    3. $h_1(M) \intersect M' = M \intersect M' \setminus\theset{p, q}$.


Definition (Homological Intersection Number)
: If $M, N \subset V$ are closed smooth submanifolds, then for $[M], [N] \subset H_*(V)$, then $[M] \cdot [N] = \sum_{p\in M\transverse N} \eps(p)$ is the *homological intersection number*.

*Sketch of proof:*

1. Given $L$, find a $D^2 \injects V$ that it bounds.
  Note that $D^2$ may have self-intersections.

2. Continuous maps can be approximated by smooth maps, and smooth intersections can be perturbed to be transverse.
  This lets the disc be perturbed, and since $2+2\leq 5$, the self-intersection can be made zero.

3. Something else.


## Facts From Differential Geometry

Let $M$ be smooth, then there exists a Riemannian metric $\inner{\wait}{\wait}$ on $T_pM$ which is symmetric and positive definite.

Given $p, q\in M$ and a curve $c(t)$ from $p$ to $q$, we want a parallel transport map $f_c: T_pM \to T_q M$.

The exponential map: something that maps a neighborhood in $T_pM$ to a neighborhood of $p$ in $M$.
Take geodesics starting at $p$ and evaluate at $t=1$.

Definition (Geodesics)
: **Geodesics** are curves of global shortest length.

Definition (Normal Bundle)
: For $M\subset V$, then $TM \subset \restrictionof{TV}{M}$ is a subbundle with a metric induced from the metric on $V$.
  The **normal bundle** is $TM\perp$.

Definition (Totally geodesic submanifold)
: If $M \subset V$ is a submanifold with $p\in M$ and $v\in T_pM$, then $M$ is **totally geodesic** iff the entire geodesic starting at $p$ with initial velocity $v$ is entirely contained in $M$.

Fact (Existence of the Levi-Cevita Connection)
: Any Riemannian metric comes with a canonical **connection**: the Levi-Cevita connection.

Parallel transport along a curve in a totally geodesic submanifold (?).

## Proving Whitney's Trick

Lemma
:   Let $L_0, L_1$ be the image of $C_0, C_0' \subset \RR^2$.
    Let $U$ be a neighborhood of $C_0 \union C_0'$ in $\RR^2$, including the region they bound:

    ![Image](figures/2020-02-18-12:21.png)\

    We can extend the maps embedding $U \intersect \qty{ C_0 \intersect C_0' }$ to $\phi_1: U \to V$ be the embedding, so $\restrictionof{\phi_1}{\bd D^2} = L$.
    We then get a map

    \begin{align*}
    \phi: U \cross \RR^{m-1} \cross \RR^{n-1} &\to V \\
    \phi\inv(M) &= \qty{ U \intersect C_0  } \cross \RR^{m-1} \cross \theset{0} \\
    \phi\inv(M') &= \qty{ U \intersect C_0'  } \cross \theset{0} \cross \RR^{m-1} \\
    .\end{align*}