Charts and Atlases, Manifolds and Smooth Structures

1. Manifolds

We will introduce the machinery necessary for defining a smooth manifold.

1.1. Charts

1. Definition. Let ${X\subset M}$ be some set. An ${n}$-dimensional Chart consists of

1. an open subset ${U\subset\mathbb{R}^{n}}$
2. a map ${\varphi\colon U\to X}$

such that ${\varphi}$ is an appropriate isomorphism (for topological manifolds, it is a homeomorphism; smooth manifolds require a diffeomorphism; and so on).

2. Remark. We call ${\varphi\colon U\to X}$ a Parametrization of ${X}$, and ${\varphi^{-1}\colon X\to U}$ a Local System of Coordinates.

3. Remark. Since ${\varphi}$ is an isomorphism, the literature mixes up using ${U\to X}$ and ${X\to U}$. Milner uses ${\varphi\colon U\to X}$, but John Lee uses the opposite convention.

4. Definition. Let ${(U,\varphi)}$, ${(V,\psi)}$ be two charts. We say they are Compatible if

1. the set ${(\varphi^{-1}\circ\psi)(V)\subset U}$ is an open set;
2. the set ${(\psi^{-1}\circ\varphi)(U)\subset V}$ is an open set;
3. the map ${\psi^{-1}\circ\varphi\colon\varphi^{-1}(\psi(V))\to\psi^{-1}(\varphi(U))}$ is smooth; and
4. the map ${\varphi^{-1}\circ\psi\colon\psi^{-1}(\varphi(U))\to\varphi^{-1}(\psi(V))}$ is smooth.

In particular, the charts are compatible if ${\varphi(U)\cap\psi(V)=\emptyset}$ is disjoint.

5. Remark. We refer to the maps ${\psi^{-1}\circ\varphi}$ as Transition Functions. The condition of smooth is ${C^{\infty}(\mathbb{R}^{n})}$, but different manifolds have different conditions (we could have ${C^{k}}$ charts, or ${C^{0}}$ charts, or analytic ${C^{\omega}}$ charts, or...).

In the older literature (e.g., Kobayashi and Nomizu's Foundations of Differential Geometry), the collection of transition functions form a gadget called a Pseudogroup.

6. Remark. We abuse notation, and could be more explicit by writing $$\psi^{-1}\circ\varphi\colon\varphi^{-1}(\varphi(U)\cap\psi(V))\to\psi^{-1}(\varphi(U)\cap\psi(V)) \tag{1}$$

▸ Exercise 1. Prove chart compatibility is an equivalence relation.

1.2. Atlases

7. Definition. Let ${M}$ be a set. An (${n}$-dimensional) Atlas consists of a collection ${\{(U_{\alpha},\varphi_{\alpha})\mid\alpha\in A\}}$ of ${n}$-dimensional charts on ${M}$ such that

1. Covers ${M}$: $\displaystyle{\bigcup_{\alpha\in A}\varphi_{\alpha}(U_{\alpha})=M}$
2. Pairwise compatible: for any ${\alpha}$, ${\beta\in A}$ the charts ${(U_{\alpha},\varphi_{\alpha})}$ and ${(U_{\beta},\varphi_{\beta})}$ are compatible.

8. Definition. Two ${n}$-dimensional atlases on ${M}$, ${\mathcal{A}}$ and ${\mathcal{B}}$, are called Equivalent if their union ${\mathcal{A}\cup\mathcal{B}}$ is also an atlas. That is to say, if any chart of ${\mathcal{A}}$ is compatible with any chart of ${\mathcal{B}}$.

9. Remark. Remember: charts are compatible, but atlases are equivalent.

10. Lemma. Let ${\mathcal{B}}$ be an atlas, let ${(U,\varphi)}$ and ${(V,\psi)}$ be two charts not contained in ${\mathcal{B}}$. If ${(U,\varphi)}$ is compatible with every chart of ${\mathcal{B}}$, and if ${(V,\psi)}$ is compatible with every chart of ${\mathcal{B}}$, then ${(U,\varphi)}$ is compatible with ${(V,\psi)}$.

11. Theorem. Equivalence of atlases is an equivalence relation.

Proof: Let ${\mathcal{A}}$, ${\mathcal{B}}$, ${\mathcal{C}}$ be arbitrary atlases on ${M}$.

1. Reflexivity: ${\mathcal{A}}$ is equivalent to itself, since by definition any pair of charts in ${\mathcal{A}}$ are compatible.
2. Symmetry: let ${\mathcal{A}}$ and ${\mathcal{B}}$ be equivalent atlases, then ${\mathcal{B}}$ and ${\mathcal{A}}$ are equivalent atlases.
3. Transitivity: this is the nontrivial part. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be equivalent atlases, and ${\mathcal{B}}$ be equivalent to ${\mathcal{C}}$. Then transitivity follows by considering arbitrary charts ${(U,\varphi)\in\mathcal{A}}$ and ${(V,\psi)\in\mathcal{C}}$, then applying Lemma 10.

Thus "equivalence of atlases" forms an equivalence relation. ∎

12. Proposition. The collection of atlases on a given set ${M}$ is a set, not a proper class.

Proof: The class of atlases is a subcollection of $$\mathcal{X}=\mathcal{P}\left(\bigcup_{U\in\mathcal{P}(\mathbb{R}^{n})}\mathop{\rm Hom}\nolimits(U,M)\right) \tag{2}$$ where ${\mathop{\rm Hom}\nolimits(U,\mathbb{R}^{n})}$ is the collection of (appropriately smooth, or continuous, or holomorphic, or...) functions from ${U}$ to ${M}$. By ZF axioms, ${\mathcal{X}}$ is a set. ∎

1.3. Manifolds

13. Definition. Let ${M}$ be a set, let ${\mathcal{A}}$ be an ${n}$-dimensional atlas on ${M}$. We call a subset ${B\subset M}$ Open (with respect to ${\mathcal{A}}$) if for any chart ${(U,\varphi)\in\mathcal{A}}$ the preimage ${\varphi^{-1}(B)}$ is open (in ${U}$, and thus open in ${\mathbb{R}^{n}}$). In particular, the images ${\varphi(U)}$ are open.

14. Theorem. If two atlases ${\mathcal{A}_{1}}$ and ${\mathcal{A}_{2}}$ on ${M}$ are equivalent, then a subset ${B\subset M}$ is open with respect to ${\mathcal{A}_{1}}$ if and only it is open with respect ${\mathcal{A}_{2}}$.

15. Remark. This theorem shows an equivalence class of atlases on ${M}$ makes ${M}$ a topological space. We may therefore meaningfully speak about topological properties of ${M}$ (like compactness, connectedness, and so forth).

16. Corollary. Let ${\mathcal{A}}$ be an ${n}$-dimensional atlas for ${M}$. Then the collection of open sets with respect to ${\mathcal{A}}$ form a topology on ${M}$.

17. Definition. Let ${M}$ be a fixed set. A ${n}$-Dimensional Differential Structure (or ${n}$-Dimensional Smooth Structure) on ${M}$ consists of an equivalence class ${\mathfrak{D}}$ of ${n}$-dimensional atlases on ${M}$ such that

1. Second-Countable: ${\mathfrak{D}}$ contains an at most countable atlas;
2. Hausdorff: for any distinct ${p,q\in M}$, there exists disjoint open neighborhoods ${U,V\subset M}$ such that ${p\in U}$ and ${q\in V}$.

18. Remark (Smooth Structure using a Maximal Atlas). Equivalence classes are awkward to work with, and so it is more popular to consider maximal atlases. An atlas ${\mathcal{A}}$ is maximal if it contains all charts compatible with every chart in ${\mathcal{A}}$. Given an equivalence class ${\mathfrak{A}}$ of atlases, we may obtain a maximal atlas by considering $$\mathcal{A}_{\text{max}} = \bigcup_{\mathcal{A}\in\mathfrak{A}}\mathcal{A}. \tag{3}$$ This may be used instead of an equivalence class of atlases in defining a differential structure, provided the second-countable axiom is reworded as: ${\mathcal{A}_{\text{max}}}$ contains an at most countable subatlas.

19. Remark (Convenient Fiction). No one actually constructs either a maximal atlas or a differential structure. We typically construct a smooth atlas on ${M}$, then announce we are working with the differential structure containing our atlas. Thus maximal atlases and, to some degree, differential structures are a convenient fiction.

20. Definition. A (Smooth) ${n}$-Dimensional Manifold consists of a set ${M}$ equipped with an ${n}$-dimensional differential structure.

21. Puzzle. Is this definition correct? By this, I mean: is an "${n}$-Dimensional Differential Structure" actually structure (in the sense of "stuff, structure, and properties")?