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.

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