A description/proof of that homeomorphic topological manifolds can have equivalent atlases
Topics
About: topological manifold
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of topological manifold.
- The reader knows a definition of atlas.
Target Context
- The reader will have a description and a proof of the proposition that any homeomorphic topological manifolds can have equivalent atlases or no atlas.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any topological manifolds, \(M_1\) and \(M_2\), that are homeomorphic to each other by \(f: M_1 \rightarrow M_2\), if \(M_1\) can have an atlas, \(M_2\) can have the equivalent atlas; if \(M_1\) can have no atlas, \(M_2\) can have no atlas.
"equivalent" there means that for any chart, \((U, \phi)\) on \(M_1\), the corresponding chart on \(M_2\) is \((f (U), \phi (f^{-1}))\).
2: Proof
\((f (U), \phi (f^{-1}))\) is a chart, because \(\phi (f^{-1})\) is a homeomorphism from f (U), open on \(M_2\) (because f is a homeomorphism), to \(\phi (U)\), open on \(\mathbb{R}^n\) (because \(\phi\) is the char map), as the compound of homeomorphisms. The set of the charts on \(M_2\) forms an \(C^\infty\) atlas, because it covers \(M_2\) (any point on \(M_2\) is f (p), so it is in f (U) where \(p \in U\)); for any area shared by \(f (U_1)\) and \(f (U_2)\), the charts transition map, \(\phi_1 (f^{-1} (f (U_1))) \rightarrow \phi_2 (f^{-1} (f (U_2)))\), is \(\phi_1 (U_1) \rightarrow \phi_2 (U_2)\), which is nothing but the charts transition map on \(M_1\), \(C^\infty\).
If \(M_1\) can have no atlas, \(M_2\) can have no atlas, because if \(M_2\) had an atlas, there would be the equivalent atlas on \(M_1\), a contradiction.
3: Note
Some people will deem this proposition obvious: "The 2 topological manifolds are the same, because they are homeomorphic!".
Well, one of my pet peeves for some prevalent discourses on topological space or manifold is that they breezily say that "2 entities are the same" just because they are equivalent. 'Being equivalent' means 'sharing a set of properties', not 'being the same'.
Even if the Earth surface and the Mars surface are homeomorphic, they are not the same: a point you are standing at is not any point on the Mars surface, even if the point homeomorphicly corresponds to a point on the Mars surface.
I have bothered to describe and prove the proposition, because I am offended by such breeziness, although some quick-witted people may deem the proposition obvious anyway. .
The breezy people will say "The topological manifolds share the same atlas.", but I say that the 2 atlas are just equivalent, not the same.
An atlas for the Earth surface is not any atlas for the Mars surface, even if a Mars surface atlas can be derived from the Earth surface atlas via the homeomorphism.
