Manifold

@lee2013smooth page 23 and 29

See also submanifold.

A topological manifold of dimension $n$ is a topological space $(M,\mathcal{O})$ with the properties:

• It is Hausdorff
• It is second countable.
• It is locally euclidean of dimension $n$. For every point $p$ in the manifold $M$, there exists a neighborhood $U\subseteq M$ containing $p$ and a homeomorphism $\varphi :U\to V$ onto an open subset $V\subseteq {\mathbb{R}}^n$. The pairs $(U,\varphi )$ are called charts.

A fundamental theorem of topology states: for $n_1,n_2\in \mathbb{N}$ (two different natural numbers, $n_1\ne n_2$), the Cartesian spaces ${\mathbb{R}}^{n_1}$ and ${\mathbb{R}}^{n_2}$ of dimension $n_1$ and $n_2$, respectively, are not homeomorphic (while for $n_1=n_2$ they are, of course). Hence, the dimension of topological manifolds is a topological invariant.

This seems intuitively obvious, but the formal proof (due to Brouwer around 1910) is comparatively hard, and the problem was an important open problem in the 19th century. To appreciate that the problem is harder than it may superficially seem, it serves to notice that for every $n\in \mathbb{N}$, there are surjective continuous maps ${\mathbb{R}}^1\to {\mathbb{R}}^n$ (the Peano curves), in addition to the obvious injective continuous maps ${\mathbb{R}}^1\to {\mathbb{R}}^n$ for $n\ge 1$ (or indeed ${\mathbb{R}}^m\to {\mathbb{R}}^n$ for $m\le n$). Because of this, there are also bijective (but discontinuous) maps ${\mathbb{R}}^m\to {\mathbb{R}}^n$ for $m,n\ge 1$ (by the Schroeder–Bernstein theorem).

Given a topological manifold we can construct a maximal atlas $A$ by adding all the possible charts.

Differentiable manifolds

Definition (Compatible Charts). Two charts $(U,x)$ and $(V,y)$ of a topological manifold $(\mathcal{M},\mathcal{O})$ are called $◻$-compatible${}^2$ if either

• (a) $U\cap V=\mathrm{\varnothing }$, or
• (b) $U\cap V\ne \mathrm{\varnothing }$ and the chart transition maps $(x\circ y^{-1}):y(U\cap V)\to x(U\cap V)$ and $(y\circ x^{-1}):x(U\cap V)\to y(U\cap V)$ have the property $◻$.

Definition (Compatible/Restricted Atlas). An atlas ${\mathcal{A}}_{◻}$ is a $◻$-compatible atlas if all of its charts are $◻$-compatible.

Definition. A $◻$-manifold is the triple $(\mathcal{M},\mathcal{O},{\mathcal{A}}_{◻})$.

Example: A smooth manifold is a topological manifold $M$ with a subatlas ${\mathcal{A}}_{inf}\subset A$ which is $C^{\mathrm{\infty }}$-compatible.

Theorem 4.2.1 (Whitney Theorem). Any $C^k$-atlas $A_{C^k}$ for $k\ge 1$ for a topological manifold, contains as a subatlas a $C^{\mathrm{\infty }}$ atlas.

Theorem 4.3.2. The number of $C^{\mathrm{\infty }}$-manifolds one can make from a given $C^0$-manifold (if any), up to diffeomorphism is given by the following table:

The first three results in the table are the so-called Moise-Radon theorems, and the 5,6,7,... results are shown using an area of topology known as surgery theory.