I am very confused by the notion of the maximal atlas.
Given some separable metrizable $C^r$ manifold $\mathcal{M}$, from what I read from the internet and the book, there might be more than one maximal atlases covering it, i.e. they are not compatible with one another, and they have different $C^r$ structures.
Can anyone give me some clear explicit example(s)? (Due to the definition of maximal atlases, of course, you just need to provide non-compatible beginning atlases rather than the maximal ones).
I would really appreciate it.
Here's the definition of compatibility from the material I read:
Suppose we have two coordinate patches in $M$, $F_i:U_i\rightarrow V_i\subset \mathbb{R}^n$, i=1,2. The coordinate patches are $C^r$-compatible (where $r\in \mathbb{N}$ or $r=\infty$) if
- Either $U_1\cap U_2= \emptyset$,
- or if $U_1\cap U_2 \neq \emptyset$ but $F_{12}:=F_1\circ F_2^{-1}: F_2(U_1\cap U_2)\leftrightarrow F_1(U_1\cap U_2)$ is a bijection of $C^r(F_2(U_1\cap U_2);\mathbb{R}^n)$ and
$F_{21}:=F_2\circ F_1^{-1}: F_1(U_1\cap U_2)\leftrightarrow F_2(U_1\cap U_2)$ is a bijection of $C^r(F_1(U_1\cap U_2);\mathbb{R}^n)$
Best Answer
Consider the real line with a single coordinate patch given by $F_1: x\mapsto |x|^{t-1}x$ where $t$ is any real number $> 1$. You can verify that $F_1$ is a homeomorphism whose inverse is not differentiable at $0$. It is incompatible with the usual differential structure given by the coordinate $F_2: x \mapsto x$, because at the point $0$ the map $F_2 \circ F_1^{-1}$ is not differentiable. This shows that there are at least uncountably many smooth structures on the real line; you can generalise the method to generate uncountably many smooth structures from any given smooth manifold.