Math4202 Topology II (Lecture 4)

Manifolds

Imbedding of Manifolds

Definition of Manifold

An $m$ -dimensional manifold is a topological space $X$ that is

Hausdorff
With a countable basis
Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$ . (local euclidean)

Note

Try to find some example that satisfies some of the properties above but not a manifold.

Non-Hausdorff
Non-countable basis
- Consider $\mathbb{R}^\delta$ where the set is $\mathbb{R}$ with discrete topology. The basis must include all singleton sets in $\mathbb{R}$ therefore $\mathbb{R}^\delta$ is not second countable.
Non-local euclidean
- Consider the subspace topology over segment $[0,1]$ on real line, the subspace topology is not local euclidean since the open set containing the end point $[0,a)$ is not homeomorphic to open sets in $\mathbb{R}$ . (if we remove the end point, in the segment space we have $(0,a)$ but in $\mathbb{R}$ is $(-a,0)\cup (0,a)$ , which is not connected. Therefore cannot be homeomorphic to open sets in $\mathbb{R}$ )
- Any shape with intersection is not local euclidean.

Whitney’s Embedding Theorem

If $X$ is a compact $m$ -manifold, then $X$ can be imbedded in $\mathbb{R}^N$ for some positive integer $N$ .

In general, $X$ is not required to be compact. And $N$ is not too big. For non compact $X$ , $N\leq 2m+1$ and for compact $X$ , $N\leq 2m$ .

Definition for partition of unity

Let $\{U_i\}_{i=1}^n$ be a finite open cover of topological space $X$ . An indexed family of continuous function $\phi_i:X\to[0,1]$ for $i=1,...,n$ is said to be a partition of unity dominated by $\{U_i\}_{i=1}^n$ if

$\operatorname{supp}(\phi_i)=\overline{\{x\in X: \phi_i(x)\neq 0\}}\subseteq U_i$ (the closure of points where $\phi_i(x)\neq 0$ is in $U_i$ ) for all $i=1,...,n$
$\sum_{i=1}^n \phi_i(x)=1$ for all $x\in X$ (partition of function to $1$ )

Existence of finite partition of unity

Let $\{U_i\}_{i=1}^n$ be a finite open cover of a normal space $X$ (Every pair of closed sets in $X$ can be separated by two open sets in $X$ ).

Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$ .

A more generalized version, If the space is paracompact, then there exists a partition of unity dominated by $\{U_i\}_{i\in I}$ with locally finite. (Theorem 41.7)

Proof for Whithney's Embedding Theorem

Since $X$ is a compact manifold, $\forall x\in X$ , there is an open neighborhood $U_x$ of $x$ such that $U_x$ is homeomorphic to $\mathbb{R}^d$ . That means there exists $\varphi_i:U_x\to \varphi(U_x)\subseteq \mathbb{R}^m$ .

Where $\{U_x\}_{x\in X}$ is an open cover of $X$ . Since $X$ is compact, there is a finite subcover $\bigcup_{i=1}^k U_{x_i}=X$ .

Apply the existsence of partition of unity, we can find a partition of unity dominated by $\{U_{x_i}\}_{i=1}^k$ . With family of functions $\phi_i:\mathbb{R}^d\to[0,1]$ .

Define $h_i:X\to \mathbb{R}^m$ by

h_i(x)=\begin{cases} \phi_i(x)\varphi_i(x) & \text{if }x=x_i\\ 0 & \text{otherwise} \end{cases}

We claim that $h_i$ is continuous using pasting lemma.

On $U_i$ , $h_i=\phi_i\varphi_i$ is product of two continuous functions therefore continuous.

On $X-\operatorname{supp}(\phi_i)$ , $h_i=0$ is continuous.

By pasting lemma, $h_i$ is continuous.

Continue on next lecture.