Topology/Manifolds

The surface of a sphere and a 2-dimensional plane, both existing in some 3-dimensional space, are examples of what one would call surfaces. A topological manifold is the generalisation of this concept of a surface. If every point in a topological space has a neighbourhood which is homeomorphic to an open subset of $\mathbb {R} ^{n}$, for some non-negative integer $n$, then the space is locally Euclidean. This formalises the idea that, while a surface might be unusually connected, patches of the surface can still resemble a euclidean space. For instance, the surface of a Klein bottle cannot be immersed in three dimensions without intersecting itself, and there is no distinction between the inside and the outside of a Klein bottle, but small patches of it still look like a euclidean space. A small creature living on the surface of a Klein bottle may not be aware of how it is connected overall or how it curves, but it could still make a rectangular or square map of its immediate area and use the map to measure lengths and directions and so forth.

A topological manifold is a locally-Euclidean Hausdorff space. Other properties are usually included in the definition of a topological space, such as being second-countable (having a countable base), which is included in the definition below. A topological manifold being Hausdorff excludes some pathological examples, such as the line with two origins, which is created by replacing the origin of the real line with two points. Any neighbourhood of either of the two origin points will contain all points in some open interval around zero, and thus will contain the other origin point, so their neighbourhoods will always intersect, so the space is not Hausdorff. See Non-Hausdorff manifold for other examples.

Definition 1 (Topological Manifold)

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

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

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 satisfy 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 maps

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. ∎