description/proof of that local diffeomorphism between oriented \(C^\infty\) manifolds with boundary is orientation-preserving or orientation-reversing over connected subspace
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of oriented \(C^\infty\) manifold with boundary.
- The reader knows a definition of orientation-preserving local diffeomorphism between oriented \(C^\infty\) manifolds with boundary.
- The reader knows a definition of orientation-reversing local diffeomorphism between oriented \(C^\infty\) manifolds with boundary.
- The reader knows a definition of connected topological space.
- The reader knows a definition of subspace topology of subset of topological space.
- The reader admits the proposition that any local diffeomorphism between any oriented \(C^\infty\) manifolds with boundary is locally orientation-preserving or orientation-reversing.
Target Context
- The reader will have a description and a proof of the proposition that any local diffeomorphism between any oriented \(C^\infty\) manifolds with boundary is orientation-preserving or orientation-reversing over each connected subspace.
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:
\(M_1\): \(\in \{\text{ the } d \text{ -dimensional oriented } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the } d \text{ -dimensional oriented } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the local diffeomorphisms }\}\)
\(T_1\): \(\in \{\text{ the connected subspaces of } M_1\}\)
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Statements:
\(\forall t \in T_1 (\forall (v_1, ..., v_d) \in \{\text{ the oriented bases for } T_tM_1\} ((d f v_1, ..., d f v_d) \in \{\text{ the oriented bases for } T_{f (t)}M_2\}))\)
\(\lor\)
\(\forall t \in T_1 (\forall (v_1, ..., v_d) \in \{\text{ the oriented bases for } T_tM_1\} ((d f v_1, ..., d f v_d) \in \{\text{ the negatively-oriented bases for } T_{f (t)}M_2\}))\)
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2: Note
Each of the definition of orientation-preserving local diffeomorphism and the definition of orientation-reversing local diffeomorphism has required that the map is orientation-preserving or orientation-reversing at each point of the domain.
Then, is each local diffeomorphism orientation-preserving or orientation-reversing?
No, because the domain may not be connected, and the map may be orientation-preserving over a connected-component and be orientation-reversing over another connected-component.
This proposition is claiming that the map is at least orientation-preserving or orientation-reversing over each connected subspace.
As each connected-component is a connected subspace by the proposition that any connected topological component is exactly any connected topological subspace that cannot be made larger, the map is orientation-preserving or orientation-reversing over each connected-component.
3: Proof
Whole Strategy: Step 1: for each \(t_1 \in T_1\), take an open neighborhood of \(t_1\), \(U_{t_1}\), such that \(f \vert_{U_{t_1}}: U_{t_1} \to f (U_{t_1})\) is orientation-preserving or orientation-reversing; Step 2: take the union of the orientation-preserving open neighborhoods, \(U\), and the union of the orientation-reversing open neighborhoods, \(U'\), and see that \(T_1 \cap U\) and \(T_1 \cap U'\) are open subsets of \(T_1\) and \((T_1 \cap U) \cap (T_1 \cap U') = \emptyset\); Step 3: see that \(T_1 \cap U = \emptyset\) or \(T_1 \cap U' = \emptyset\).
Step 1:
For each \(t_1 \in T_1\), let us take an open neighborhood of \(t_1\), \(U_{t_1} \subseteq M_1\), such that \(f \vert_{U_{t_1}}: U_{t_1} \to f (U_{t_1})\) is orientation-preserving or orientation-reversing, which is possible by the proposition that any local diffeomorphism between any oriented \(C^\infty\) manifolds with boundary is locally orientation-preserving or orientation-reversing.
Step 2:
Let us take the union of the orientation-preserving open neighborhoods taken in Step 1, \(U\).
Let us take the union of the orientation-reversing open neighborhoods taken in Step 1, \(U'\).
\(U\) and \(U'\) are open on \(M_1\) as some unions of open subsets.
So, \(T_1 \cap U\) and \(T_1 \cap U'\) are open subsets of \(T_1\), by a definition of subspace topology of subset of topological space.
\((T_1 \cap U) \cap (T_1 \cap U') = \emptyset\), because if \(u \in (T_1 \cap U) \cap (T_1 \cap U')\), \(f\) would be orientation-preserving and orientation-reversing at \(u\), which would be impossible because being orientation-preserving at a point and being orientation-reversing at the same point are exclusive: for any oriented basis for \(T_uM_1\), \((b_1, ..., b_d)\), if \((d f_u b_1, ..., d f_u b_d)\) is oriented, \(f\) is orientation-preserving at \(u\) but not orientation-reversing, while if \((d f_u b_1, ..., d f_u b_d)\) is negatively-oriented, \(f\) is orientation-reversing but not orientation-preserving.
Step 3:
\(T_1 = (T_1 \cap U) \cup (T_1 \cap U')\), because each \(t_1 \in T_1\) is in \(U\) or \(U'\).
If \(T_1 \cap U \neq \emptyset\) and \(T_1 \cap U' \neq \emptyset\), \(T_1\) would not be connected, a contradiction against that \(T_1\) is a connected subspace.
So, \(T_1 \cap U = \emptyset\) or \(T_1 \cap U' = \emptyset\).
That means that all the open neighborhoods taken in Step 1 are orientation-preserving or all the open neighborhoods taken in Step 1 are orientation-reversing.
That obviously implies this proposition.
