Differential Topology Lecture 1 Notes

This is the first of a series of lecture notes. I am going to try to keep this up so that it forces me to review the notes I have taken in lectures. Unfortunately, I cannot guarantee these will be any good.

Highlights of the Course

1. The first interesting theorem we will cover is Whitney’s Theorem: Every $n$-dimensional smooth manifold can be embedded in $\mathbb{R}^{2n+1}$.

An embedding is an injective map with no self intersections.

2. Jordon Curve Theorem: Every simple closed curve in $\mathbb{R}^2$ splits $\mathbb{R}^2$ into connected components.

Jordon-Brower Theorem is the generalization that a simple closed $n-1$ dimensional manifold splits $\mathbb{R}^n$.

3. Euler Characteristic of a manifold $M$, $\chi(M)$ ($\chi = V - E + F$). For example $\chi(\mathbb{R}) = 1$, $\chi(S^2) = 2$, $\chi(T^2) = 0$.

Hedgehog Theorem says $S^2$ cannot be combed. This is the same as saying there is always a still point or there is always a point with no wind on earth.

A compact manifold $M$ can be combed if and only if $\chi(M) = 0$, e.g. $T^2$ can be combed.

Fixed point problems. Then $f : M \to M$ a smooth deformation of the identity, if $\chi(M) \neq 0$, then $f$ has at least one fixed point. A circle rotated by $\theta$ has no fixed points.

Topological Manifolds

Definition An $n$-dimensional topological manifold, $M$, is a topological space such that:

1. $M$ is Hausdorff,
2. $M$ is second-countable,
3. $M$ is locally euclidean.

Definition A chart on $M$ is a pair $(U,f)$, $U \subseteq M$ such that $f : U \to V \subseteq \mathbb{R}^n$ and $V$ is open.

Example Two rays in $\mathbb{R}^2$ shooting out from the origin. Let $f$ be the projection of these two lines to $\mathbb{R}$.

Examples of spaces that are not manifolds.

Properties of Topological Manifolds

1. Locally euclidean $\implies$ locally path connected $\implies$ disjoint union of path connected manifolds.

Topologist’s Sine Curve is connected but not path connected.

2. They are locally compact

3. They are paracompact.

Definition $M$ is locally compact iff $\forall x \in M$, $\exists C \subseteq M$ and $C$ compact such that $C \supsetneq N^{\textrm{open}}(x)$.

Definition A cover $\mathscr{B}$ of a space $M$ is a refinement of $\mathscr{A}$ iff $\forall B \in \mathscr{B}$, $\exists A \in \mathscr{A}$ where $B \subseteq A$.

Definition $M$ is locally finite iff $\forall x \in M$, $\exists U \subseteq M$ open such that only finite members of an open cover $\mathscr{B}$ intersect $U$ nonemptily.

Definition $M$ is paracompact iff every open cover $\mathscr{A}$ has a locally finite refinement.