# Topology/Manifolds

Topology
 ← Čech Homology Manifolds Manifolds/Categories of Manifolds →

Definition 1 (Topological Manifold)

A topological space $M$ is called an $n$-dimensional topological manifold (or $n$-manifold) if,

1. $M$ is Hausdorff.
2. $M$ is second-countable.
3. Every point $x \in M$ has an open neighbourhood $U \subset M$ that is homeomorphic to an open subset of $\mathbb{R}^n$.

Note: As a convention, the ball $B^0$ is a single point. Any space with the discrete topology is a 0-dimensional manifold.

Note also that all topological manifolds are clearly locally connected.

To emphasize that a given manifold $M$ is $n$-dimensional, we will use the shorthand $M^n$. This is not to be confused with an $n$-ary cartesian product $M\times...\times M$. However, we will prove later that such a construction does exist as well.

The alert reader may wonder why we require the manifold to be Hausdorff and second-countable. The reason for this is to exclude some pathological examples. Two such examples are the long line, which is not second-countable, and the line with two origins, which is not Hausdorff.

Theorem 2

A topological manifold is connected if and only if it is pathwise connected.

Proof: Since all topological manifolds are clearly locally connected, the theorem immediately follows.

Definition 3

Let $M$ be a topological $n$-manifold. Let $p\in M$ and $p\in U\subseteq M$ be an open neighborhood of $p$. Now, let $\phi\,:\, U \rightarrow U^\prime$, where $U^\prime \subseteq \mathbb{R}^n$, be a homeomorphism. Then the pair $(U,\phi)$ is called a chart at $p$.

Definition 4

Let $M$ be a topological $n$-manifold, and let $A=\{(U_i,\phi_i)\}_{i\in I}$ be charts on $M$ such that $\bigcup_{i\in I} U_i = M$. That is, $\{U_i\}_{i\in I}$ is an open covering of $M$. Then $A$ is called an atlas on $M$.

Definition 5

Let $M$ be an $n$-manifold and let $(U,\phi)$ and $(V,\psi)$ be charts are a point $p$ (so $U\cap V \neq \emptyset$). Define the transition function (or chart transformation) between the two charts as the homeomorphism $\psi\circ \phi^{-1} : \phi(U\cap V) \rightarrow \psi(U\cap V)$.

Given a pair $(M,A)$, where $M$ is an $n$-manifold and $A$ is an atlas on $M$, properties that $M$ may satifsy are often expressed as properties of the transition functions between charts in $A$. This is how we will define our notion of a differentiable manifold.

Definition 6

An atlas for a manifold is smooth (or $\mathcal{C}^\infty$) if all the transition functions are smooth (All higher order partial derivatives exist and are continuous).

Definition 7

A diffeomorphism is a smooth homeomorphism $f$ such that $f^{-1}$ is also smooth.

Note that in a smooth atlas, all transition functions are diffeomorphisms.

Definition 8

Let $M$ be a manifold and $A$ be a smooth atlas on $M$. Then, define $A_{\mathrm{max}}$ as the set of all charts $(V,\psi)$ on $M$ such that for all $(U,\phi)\in A$, $\psi\circ \phi^{-1}|_{\phi(U\cap V)}$ and $\psi\circ \phi^{-1}|_{\phi(U\cap V)}$ are smooth.

A chart $(V,\psi)$ with the property described above is said to be compatible with $A$.

Lemma 9

$A_{max}$ is a smooth atlas on $M$.

Proof: We have to show that the transition functions between any pair of charts in $A_{\mathrm{max}}$ are smooth. This is obvious if one of then is in$A$, so let $(V_1,\psi_1)$ and $(V_2,\psi_2)$ be charts in $A_{\mathrm{max}}$ that are not in $A$, such that $V_1\cap V_2\neq \emptyset$. Let $(U,\phi)\in A$ be a chart such that $U\cap V_1 \cap V_2 = W\neq \emptyset$. Then $\phi\circ \psi_1^{-1}|_W$ and $\psi_2\circ \phi^{-1}|_W$ are both smooth, since, both $(V_1,\psi_1)$ and $(V_2,\psi_2)$ are compatible with $A$. Then, $\psi_2\circ \psi_1^{-1}|_W=\psi_2\circ (\phi^{-1}\circ\phi )\circ \psi_1^{-1}|_W=(\psi_2\circ \phi^{-1}|_W) \circ (\phi\circ \psi_1^{-1}|_W)$ is smooth since it is a composition of smooth maps. An identical argument for $\psi_1\circ \psi_2^{-1}$ completes the proof.

It should be obvious that if $A^\prime$ is a smooth atlas containing a smooth atlas $A$, then $A_{\mathrm{max}}^\prime=A_{\mathrm{max}}$.

## Smooth mapsEdit

Definition 10

Let $M^m$ $N^n$ be smooth manifolds, $p\in M$, and let $f\,:\, M\rightarrow N$ be a function. Then, if for any charts $(U,\phi)$ on $M$ and $(V,\psi)$ on $N$ such that $p\in U$ and $f(p)\in V$, the function $\hat{f}(\phi(p))=(\psi\circ f\circ \phi^{-1})(\psi(p))$ from $\phi(U\cap f^{-1}(V))\subseteq \mathbb{R}^m$ to $\psi(f(U)\cap V)\subseteq \mathbb{R}^n$ is a smooth function on euclidean spaces, then $f\,:\, M\rightarrow N$ is said to be smooth at $p$. $f$ is called a smooth function if it is smooth for all $p\in M$.

Lemma 11

$f\,:\, M\rightarrow N$ is smooth at $p\in M$ if and only if there exists charts $(U,\phi)$ on $M$ and $(V,\psi)$ on $N$ with $p\in U\subseteq f^{-1}(V)$ such that $(\psi\circ f\circ\phi^{-1}):\phi(U\cap f^{-1}(V)) \rightarrow \psi(f(U)\cap V)$ is smooth.

Proof: $f$ is continuous since $\psi\circ f\circ\phi^{-1}$ is smooth and thus continuous and $\phi$ and $\psi$ are homeomorphisms. Let $(U^\prime,\phi^\prime)$ and $(V^\prime,\psi^\prime)$ be two other charts at $p$ and $f(p)$. Then $(\psi^\prime\circ f\circ{\phi^\prime}^{-1})=(\psi^\prime\circ\psi^{-1})\circ (\psi\circ f \circ \phi^{-1})\circ (\phi\circ {\phi^\prime}^{-1})$ which is a composition of smooth functions since the atlases on $M$ and $N$ are smooth, and is therefore smooth.

Remark 12

By Lemma 11, we do not have to check all charts to see if a function is smooth. A relief, since maximal atlases tend to be uncountably big.

Definition 12

If $f\,:\,M\rightarrow N$ is a smooth bijective function such that its inverse is smooth too, it is called a diffeomorphism. Two manifolds are called diffeomorphic if there exists a diffeomeorphism between them.

Lemma 13

Let $f\,:\,M\rightarrow N$ and $g\,:\,N\rightarrow P$ be smooth. Then $g\circ f \,:\, M\rightarrow P$ is smooth as well.

Proof: Let $(U,\phi)$, $(V,\psi)$ and $(W,\xi)$ be charts on $M,N,P$ at $p,f(p),g\circ f(p)$ respectively. Then $\xi \circ g\circ f\circ \phi^{-1}(\phi(p))=(\xi\circ g\circ\psi^{-1})\circ (\psi\circ f\circ\phi^{-1})(\phi(p))$ which is a composition of smooth maps of euclidean space and is hence smooth.

Topology
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