I am trying to underestand the discussion at the wiki page https://en.wikipedia.org/wiki/Local_diffeomorphism#Discussion.
Let me recall the definition first:
A local diffeomorphism $f: X\rightarrow Y$ is a mapping that satisfies the following property:
For all points $x\in X$, there exists an open set $U\ni x$ such that $f(U)$ is an open set in $Y$ and the map $f\vert_{U}:U\rightarrow f(U)$ is a diffeomorphism.
In this sentence:
- Although local diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire (smooth) manifold.
1.a) What does the author mean by local differentiable structure? We do not know what differentiable structure a priori they're endowed with. Also, I think the author meant some family of mappings (not yet local diffeos) that could be glued together and promoted to a local diffeomorphism.
So to advance this argument we need to assume that there exist some smooth structures on $S^{2}$ and $\mathbb{R}^2$ with their corresponding atlases such that the chart representations of some local mappings $g: X\supset U \rightarrow g(U) \subset Y$ are indeed diffeomorphisms between open subsets in $\mathbb{R}^2$.
1.b) Is it correct to say that this differential structure was not mentioned because there is only one differential structure (up to a smooth diffeomorphism on these manifolds) for manifolds with dimensions $<4$?
1.c) Assuming such mappings exists, we can only then ask whether they can be glued together to form a local diffeomorphism, so a mapping $f:S^{2} \rightarrow \mathbb{R}^2$ giving $f\vert_{U}=g$ wherever $g$ is defined on an open set $U$.
- For example, there can be no local diffeomorphism from the 2-sphere to Euclidean 2-space although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous, the continuous image of a compact space is compact, the sphere is compact whereas Euclidean 2-space is not.
2.a) The argument does not convince me. Supposing by contradiction that a local diffeomorphism exists, it would necessarily be a continuous map $f:S_{2} \rightarrow \mathbb{R}^{2}$. We know that continuous maps preserve compactness, but this is not an obstruction to local diffeomorphism existence here, as we do not require the whole $\mathbb{R}^{2}$ to be the image of $f$. Thus, I do not see a contradiction.
Please give me your thoughts on my reasoning and correct me.
Side question:
I have once seen in more recent literature (think Springer, nicely typed Graduate Series in Mathematics) a very nice exercise on comparing differentiable structures on the real line. The exercise involved mappings of the sort $x\longmapsto x^3$ and had three suggestions on the potential chart maps denoted by a), b), c). The reader was asked to check whether the atlases are equivalent or to perform some similar task. If you knew the book I'm reffering to I'd be much obliged to learn its name.