Manifolds

Topological Manifold

An $n$-dimensional manifold, or $n$-manifold for short, is a Hausdorff and paracompact(or equivalently Hausdorff and second-countable) topological space $M$ with the property that each point has a neighbourhood that is homeomorphic to an open subset of $n$-dimensional Euclidean space ${\mathbb{R}}^n$.

Clearly, on a manifold, there might be different collections of homeomorphisms that makes it locally Euclidean. So we introduce:

Chart and Atlas

A chart for a manifold $M$ is a homeomorphism $\phi :U\to V$ where $U$ is open in $M$ and $V$ is open in ${\mathbb{R}}^n$. A collection of charts $\mathcal{A}=\{{\phi }_{\alpha }:U_{\alpha }\to V_{\alpha }|\alpha \in I\}$ is called an atlas for $M$ if ${\cup }_{\alpha \in I}{U}_{\alpha }=M$.

Compatible Chart

Two charts $\phi$ and $\eta$ are smooth compatible if $\phi \circ {\eta }^{-1}$ is a diffeomorphism ($C^{\infty }$)

e.g.

• Consider the set $M=\mathbb{R}$ with the usual topology. $\mathcal{A}=\{x\mapsto x:\mathbb{R}\to \mathbb{R}\}$ and $\mathcal{B}=\{x\mapsto x^3:\mathbb{R}\to \mathbb{R}\}$ are both differentiable atlases. They are not compatible since the union $\{x\mapsto x,x\mapsto x^3\}$ is not differentiable.
• The sphere $S^n=\{x\in {\mathbb{R}}^{n+1}:\Vert x\Vert =1\}$ is a closed subset of Euclidean space, thus the topological requirements are satisfied. Define the following two maps: ${\phi }_{+}:S^n\setminus \{N\}\to {\mathbb{R}}^n,{\phi }_{-}:S^n\setminus \{S\}\to {\mathbb{R}}^n$as follows: We write $x\in {\mathbb{R}}^{n+1}$ as $(y,z)$ where $y\in {\mathbb{R}}^n$ and $z\in \mathbb{R}$, we take ${\phi }_{+}(y,z)=\frac{y}{1-z}$ and ${\phi }_{-}(y,z)=\frac{y}{1+z}$. Then we have

Differentiable (Smooth) Atlas

Let $M$ be an $n$-dimensional manifold. An atlas $\mathcal{A}=\{{\phi }_{\alpha }:U_{\alpha }\to V_{\alpha }|\alpha \in I\}$ for $M$ is differentiable or smooth if for every $\alpha$ and $\beta$ in $I$, the map ${\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}$ is smooth, thus a diffeomorphism, as a map between open subsets of ${\mathbb{R}}^n$.

Compatible Atlases

Two differentiable atlases $\mathcal{A}$ and $\mathcal{B}$ are compatible if their union is also a differentiable atlas.

Partial Order on Smooth Atlases

Let $M$ be a topological manifold. Let $\mathcal{M}$ denote the collection of all smooth ($C^{\infty }$-compatible) atlases on $M$. We define a binary relation, denoted as a partial order $\prec$, on $\mathcal{M}$ as follows: We say that an atlas ${\mathcal{A}}_1\in \mathcal{M}$ is refined by (or is a refinement of) an atlas ${\mathcal{A}}_2\in \mathcal{M}$, denoted by ${\mathcal{A}}_1\prec {\mathcal{A}}_2,$ if for every local coordinate chart $(U_1,{\phi }_1)\in {\mathcal{A}}_1$, there exists an open cover of $U_1$ given by a subcollection of charts in ${\mathcal{A}}_2$. That is, there exist charts $\{(U_{\alpha ,(2)},{\phi }_{\alpha ,(2)}){\}}_{\alpha }\subseteq {\mathcal{A}}_2$ such that: $U_1={\bigcup }_{\alpha }{U}_{\alpha ,(2)},$ subject to the condition that the original chart $(U_1,{\phi }_1)$ is smoothly compatible ($C^{\infty }$-compatible) with every overlapping chart $(U_{\alpha ,(2)},{\phi }_{\alpha ,(2)})\in {\mathcal{A}}_2$. Under this definition, we say that ${\mathcal{A}}_2$ is a refinement of ${\mathcal{A}}_1$.

Proposition

For $M$ a topological manifold, every smooth atlas for $M$ is contained in a unique maximal smooth atlas.

Proposition

Two differentiable atlases $\mathcal{A}$ and $\mathcal{B}$ are compatible if and only if for every chart $\varphi$ in $\mathcal{A}$ and $\eta$ in $\mathcal{B}$, both $\varphi \circ {\eta }^{-1}$ and $\eta \circ {\varphi }^{-1}$ are smooth.

Smooth Manifold

A $C^k$ structure on a manifold $M$ is an equivalence class of differentiable atlases, where two atlases are deemed equivalent if they are compatible. A smooth manifold is a manifold $M$ together with a smooth ($C^{\infty }$) structure on $M$.

Remark

A given topological space can carry many different differentiable structures: For example, if $M=\mathbb{R}$ we can take two charts ${\varphi }_1(x)=x$ and ${\varphi }_2(x)=x^3$. Then both ${\varphi }_1$ and $\{{\varphi }_2\}$ are smooth atlases for $M$, but they define different smooth structures.

e.g. $\{v\in {\mathbb{R}}^n\mid v\}$ is a smooth manifold.

Theorem

${\mathbb{R}}^n$ for all $n\ne 4$ has a unique smooth structure. ${\mathbb{R}}^4$ has uncountably many smooth structures.

See more: An Application of Gauge Theory to Four Dimensional Topology Simon Donaldson