description/proof of that for 2-sheeted \(C^\infty\) covering map from oriented \(C^\infty\) manifold with boundary onto connected \(C^\infty\) manifold with boundary, for every evenly-covered trivializing open subset of codomain, 2 restrictions of map induce uniformly same or different orientations on open subset
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) covering map.
- The reader knows a definition of \(C^\infty\) trivializing open subset and \(C^\infty\) local trivialization.
- The reader knows a definition of orientation of \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset.
- The reader admits the proposition that any local diffeomorphism between any oriented \(C^\infty\) manifolds with boundary is orientation-preserving or orientation-reversing over each connected subspace.
Target Context
- The reader will have a description and a proof of the proposition that for any 2-sheeted \(C^\infty\) covering map from any oriented \(C^\infty\) manifold with boundary onto any connected \(C^\infty\) manifold with boundary, for every evenly-covered trivializing open subset of the codomain, the 2 restrictions of the map induce uniformly same or different orientations on the open subset.
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:
\(M_1\): \(\in \{\text{ the oriented } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the connected } d \text{ -dimensional } C^\infty \text{ manifolds with boundary }\}\)
\(\pi\): \(: M_1 \to M_2\), \(\in \{\text{ the 2-sheeted } C^\infty \text{ covering maps }\}\)
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Statements:
\(\forall U \in \{\text{ the evenly-covered open subsets of } M_2\} \cap \{\text{ the } C^\infty \text{ trivializing open subsets of } M_2\} (\text{ the orientation induced by } \pi \vert_{{\pi}^{-1} (U)_1} = \text{ the orientation induced by } \pi \vert_{{\pi}^{-1} (U)_2})\), where \({\pi}^{-1} (U)_1\) and \({\pi}^{-1} (U)_2\) denote the 2 sheets of \(U\)
\(\lor\)
\(\forall U \in \{\text{ the evenly-covered open subsets of } M_2\} \cap \{\text{ the } C^\infty \text{ trivializing open subsets of } M_2\} (\text{ the orientation induced by } \pi \vert_{{\pi}^{-1} (U)_1} \neq \text{ the orientation induced by } \pi \vert_{{\pi}^{-1} (U)_2})\), where \({\pi}^{-1} (U)_1\) and \({\pi}^{-1} (U)_2\) denote the 2 sheets of \(U\)
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2: Note
As an immediate corollary, when \(M_2\) is non-orientable, each pair of restrictions induce some different orientations, because otherwise, \(\pi\) would induce a consistent orientation on \(M_2\) from \(M_1\): each point on \(M_2\) could be given the induced orientation and each point would be on the evenly-covered trivializing open subset on which the orientations were consistent.
3: Proof
Whole Strategy: Step 1: take any open cover of \(M_2\) with some evenly-covered trivializing open subsets, \(\{U_{m_j}\}\); Step 2: classify \(\{U_{m_j}\}\) into 2 sets: the ones that the 2 restrictions of \(\pi\) induce a same orientation, \(\{V_{m_j}\}\), and the ones that the 2 restrictions of \(\pi\) induce some different orientations, \(\{W_{m_l}\}\), and see that 1 of them is empty; Step 3: suppose that \(\{W_{m_l}\}\) is empty, and for each evenly-covered trivializing \(U\), see that \(\{V_{m_j}\} \cup \{U\}\) is an open cover of \(M_2\) with evenly-covered trivializing open subsets, and see that the 2 restrictions of \(\pi\) for \(U\) induce a same orientation; suppose that \(\{V_{m_j}\}\) is empty, and for each evenly-covered trivializing \(U\), see that \(\{W_{m_l}\} \cup \{U\}\) is an open cover of \(M_2\) with evenly-covered trivializing open subsets, and see that the 2 restrictions of \(\pi\) for \(U\) induce some different orientations.
Step 1:
Let us take any open cover of \(M_2\) with some evenly-covered trivializing open subsets, \(\{U_{m_j}\}\), which is possible because around each \(m \in M_2\), there are an evenly-covered open subset and a trivializing open subset, and the intersection of them can be taken to be \(U_m\), and \(\{U_m \vert m \in M_2\}\) can be possibly reduced to still cover \(M_2\).
\(\pi^{-1} (U_{m_j})\) consists of the 2 connected components, \(\pi^{-1} (U_{m_j})_1\) and \(\pi^{-1} (U_{m_j})_2\).
\(\pi \vert_{\pi^{-1} (U_{m_j})_1}: \pi^{-1} (U_{m_j})_1 \to U_{m_j}\) and \(\pi \vert_{\pi^{-1} (U_{m_j})_2}: \pi^{-1} (U_{m_j})_2 \to U_{m_j}\) are diffeomorphisms.
\(\pi \vert_{\pi^{-1} (U_{m_j})_1}\) induces the orientation on \(U_{m_j}\) from the orientation on \(\pi^{-1} (U_{m_j})_1\); \(\pi \vert_{\pi^{-1} (U_{m_j})_2}\) induces the orientation on \(U_{m_j}\) from the orientation on \(\pi^{-1} (U_{m_j})_2\).
As \(U_{m_j}\) is a trivializing open subset, there is a \(C^\infty\) frame over it, by the proposition that for any \(C^\infty\) vectors bundle, any \(C^\infty\) frame exists over and only over any trivializing open subset, which determines a consistent orientation over \(U_{m_j}\). So, let us give \(U_{m_j}\) the orientation, \(o\).
Then, \(\pi \vert_{\pi^{-1} (U_{m_j})_1}\) is orientation-preserving or orientation-reversing, by the proposition that any local diffeomorphism between any oriented \(C^\infty\) manifolds with boundary is orientation-preserving or orientation-reversing over each connected subspace.
That means that the orientation induced by \(\pi \vert_{\pi^{-1} (U_{m_j})_1}\) on \(U_{m_j}\) is \(o\) or \(- o\).
Likewise, the orientation induced by \(\pi \vert_{\pi^{-1} (U_{m_j})_2}\) on \(U_{m_j}\) is \(o\) or \(- o\).
Step 2:
Let us classify \(\{U_{m_j}\}\) into 2 sets: the ones that the 2 restrictions of \(\pi\) induce the same orientation, \(\{V_{m_j}\}\), and the ones that the 2 restrictions of \(\pi\) induce the different orientations, \(\{W_{m_l}\}\).
Let us suppose that none of them was empty.
There would be the pair of a \(V_{m_j}\) and a \(W_{m_l}\) such that \(V_{m_j} \cap W_{m_l} \neq \emptyset\): otherwise, \(\cup \{V_{m_j}\} \cap \cup \{W_{m_l}\} = \emptyset\) (because otherwise, \(p \in \cup \{V_{m_j}\} \cap \cup \{W_{m_l}\}\), which would imply that \(p \in V_{m_j}\) and \(p \in W_{m_l}\) and \(p \in V_{m_j} \cap W_{m_l}\)), \(\cup \{V_{m_j}\}\) and \(\cup \{W_{m_l}\}\) would be open subsets of \(M_2\), and \(\cup \{V_{m_j}\} \cup \cup \{W_{m_l}\} = M_2\), but \(\cup \{V_{m_j}\} \neq \emptyset\) and \(\cup \{W_{m_l}\} \neq \emptyset\) by the supposition, which would mean that \(M_2\) was not connected, a contradiction.
Let \(p \in V_{m_j} \cap W_{m_l}\) be any. \(\pi^{-1} (p)\) would have some 2 points, \(p_1 \in \pi^{-1} (V_{m_j})_1\) and \(p_2 \in \pi^{-1} (V_{m_j})_2\). \(p_1 \in \pi^{-1} (W_{m_l})_1\) and \(p_2 \in \pi^{-1} (W_{m_l})_2\) without loss of generality. Let the orientation induced from \(p_1\) on \(p\) be \(O\). The orientation induced from \(p_2\) on \(p\) would be \(O\) (by the nature of \(V_{m_j}\)) and \(- O\) (by the nature of \(W_{m_l}\)), a contradiction.
So, one of \(\{V_{m_j}\}\) and \(\{W_{m_l}\}\) is empty.
Step 3:
Let us suppose that \(\{W_{m_l}\}\) is empty.
Let \(U \subseteq M_2\) be any evenly-covered trivializing open subset.
\(\{V_{m_j}\} \cup \{U\}\) is an open cover of \(M_2\) with evenly-covered trivializing open subsets.
By the result of Step 2, \(\pi \vert_{{\pi}^{-1} (U)_1}\) and \(\pi \vert_{{\pi}^{-1} (U)_1}\) induce a same orientation on \(U\): otherwise, \(\{V_{m_j}\} \cup \{U\}\) would be classified into nonempty \(\{V_{m_j}\}\) and \(\{U\}\), a contradiction.
Let us suppose that \(\{V_{m_j}\}\) is empty.
Let \(U \subseteq M_2\) be any evenly-covered trivializing open subset.
\(\{W_{m_l}\} \cup \{U\}\) is an open cover of \(M_2\) with evenly-covered trivializing open subsets.
By the result of Step 2, \(\pi \vert_{{\pi}^{-1} (U)_1}\) and \(\pi \vert_{{\pi}^{-1} (U)_1}\) induce some different orientations on \(U\): otherwise, \(\{W_{m_l}\} \cup \{U\}\) would be classified into nonempty \(\{W_{m_l}\}\) and \(\{U\}\), a contradiction.
