description/proof of that for set and 2 topology-atlas pairs, iff there is common chart domains open cover and each transition is diffeomorphism, pairs are same
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any set and any 2 topologies for the set, iff there is a common open cover and each open subset of each element of the cover in one topology is open in the other topology and vice versa, the topologies are the same.
Target Context
- The reader will have a description and a proof of the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.
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: Structured Description
Here is the rules of Structured Description.
Entities:
\(S\): \(\in \{\text{ the sets }\}\)
\(O_1\): \(\in \{\text{ the topologies for } S\}\)
\(O_2\): \(\in \{\text{ the topologies for } S\}\)
\(A_1\): \(\in \{\text{ the } C^\infty \text{ -manifold-with-boundary atlases for } S\}\)
\(A_2\): \(\in \{\text{ the } C^\infty \text{ -manifold-with-boundary atlases for } S\}\)
\((O_1, A_1)\):
\((O_2, A_2)\):
//
Statements:
(
\(\exists \{U_\beta \vert \beta \in B\} \in \{\text{ the chart domains open covers in } (O_1, A_1)\} \cap \{\text{ the chart domains open covers in } (O_2, A_2)\}\)
\(\land\)
\(\forall \beta \in B\)
(
\(\phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1}: \phi_{1, \beta} (U_\beta) \to \phi_{2, \beta} (U_\beta) \in \{\text{ the diffeomorphisms }\}\), where \((U_\beta \subseteq S, \phi_{1, \beta})\) and \((U_\beta \subseteq S, \phi_{2, \beta})\) are some charts of \((O_1, A_1)\) and \((O_2, A_2)\)
)
)
\(\iff\)
\((O_1, A_1) = (O_2, A_2)\)
//
2: Note
The immediate purpose of this proposition is to confirm that 2 possible pairs constructed in a certain way with some freedom are the same, which means that the construction is unique regardless of the freedom.
3: Proof
Whole Strategy: Step 1: suppose that there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, take a chart of \((O_1, A_1)\), \((U_\beta \subseteq S, \phi_{1, \beta})\), and a chart of \((O_2, A_2)\), \((U_\beta \subseteq S, \phi_{2, \beta})\), and take \(\phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1}: \phi_{1, \beta} (U_\beta) \to \phi_{2, \beta} (U_\beta)\), diffeomorphic; Step 2: take any open subset of \(U_\beta\) in \(O_1\), \(U_1 \subseteq U_\beta\), and see that \(U_1 = {\phi_{2, \beta}}^{-1} \circ \phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1} (\phi_{1, \beta} (U_1))\) is an open subset of \(U_\beta\) in \(O_2\); Step 3: conclude that \(O_1 = O_2\); Step 4: conclude that \(A_1 = A_2\); Step 5: suppose that \((O_1, A_1) = (O_2, A_2)\), conclude that there is a common chart domains open cover and the transition for each common chart is a diffeomorphism.
Step 1:
Let us suppose that there is a common chart domains open cover and the transition for each common chart is a diffeomorphism.
Let us take a chart of \((O_1, A_1)\), \((U_\beta \subseteq S, \phi_{1, \beta})\), and a chart of \((O_2, A_2)\), \((U_\beta \subseteq S, \phi_{2, \beta})\).
\(\phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1}: \phi_{1, \beta} (U_\beta) \to \phi_{2, \beta} (U_\beta)\) is diffeomorphic, because that is what the supposition means.
Step 2:
Let us take any open subset of \(U_\beta\) in \(O_1\), \(U_1 \subseteq U_\beta\).
\(U_1 = {\phi_{2, \beta}}^{-1} \circ \phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1} (\phi_{1, \beta} (U_1))\).
\(\phi_{1, \beta} (U_1)\) is open on \(\phi_{1, \beta} (U_\beta)\), because \(\phi_{1, \beta}\) is homeomorphic. As \(\phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1}\) is diffeomorphic, it is homeomorphic, so, \(\phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1} (\phi_{1, \beta} (U_1))\) is open on \(\phi_{2, \beta} (U_\beta)\). Then, \(U_1 = {\phi_{2, \beta}}^{-1} \circ \phi_{2, \beta} \circ {\phi_{1, \beta}}^{-1} (\phi_{1, \beta} (U_1))\) is open on \(U_\beta\) in \(O_2\), because \(\phi_{2, \beta}\) is homeomorphic.
Step 3:
By the symmetry, any open subset of \(U_\beta\) in \(O_2\) is open in \(O_1\).
By the proposition that for any set and any 2 topologies for the set, iff there is a common open cover and each open subset of each element of the cover in one topology is open in the other topology and vice versa, the topologies are the same, \(O_1 = O_2\).
Step 4:
\((U_\beta \subseteq S, \phi_{2, \beta})\) is \(C^\infty\) compatible with \((U_\beta \subseteq S, \phi_{1, \beta})\), because the transition is \(C^\infty\), so, \((U_\beta \subseteq S, \phi_{2, \beta}) \in A_1\).
As \(\{(U_\beta \subseteq S, \phi_{2, \beta}) \vert \beta \in B\}\) is an atlas, which both \(A_1\) and \(A_2\) contain, \(A_1 = A_2\): the maximal atlas that contains any atlas is unique.
Step 5:
Let us suppose that \((O_1, A_1) = (O_2, A_2)\).
There is a common chart domains open cover.
The transition for each common chart is a diffeomorphism.
