description/proof of that for \(C^\infty\) covering map between \(C^\infty\) manifolds with boundary, pullback of \(C^\infty\) partition of unity subordinate to evenly-covered charts cover over codomain is \(C^\infty\) partition of unity subordinate to charts cover over domain
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 partition of unity subordinate to open cover of topological space.
- The reader knows a definition of pullback of \(q\)-covectors at point by \(C^\infty\) map between \(C^\infty\) manifolds with boundary.
- The reader knows a definition of \(r\)-open-ball chart around point on \(C^\infty\) manifold with boundary.
- The reader knows a definition of \(r\)-open-half-ball chart around point on \(C^\infty\) manifold with boundary.
- The reader admits the proposition that for any covering map, the cardinalities of the sheets are the same.
- The reader admits the proposition that for any \(C^\infty\) manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is a chart map if and only if it is a diffeomorphism.
- The reader admits the proposition that for any maps between any arbitrary subsets of any \(C^\infty\) manifolds with boundary \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
- The reader admits the proposition that for any topological space, any closed subset, and any disjoint set of open subsets whose union contains the closed subset, the intersection of the closed subset and each open subset is closed on the space.
- The reader admits the proposition that for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset.
Target Context
- The reader will have a description and a proof of the proposition that for any \(C^\infty\) covering map between any \(C^\infty\) manifolds with boundary, the pullback of any \(C^\infty\) partition of unity subordinate to any evenly-covered charts cover over the codomain is a \(C^\infty\) partition of unity subordinate to an induced charts cover over the domain.
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:
\(M_1\): \(\in \{\text{ the connected and locally path-connected } C^\infty \text{ manifolds with boundary }\}\)
\(M_2\): \(\in \{\text{ the connected and locally path-connected } C^\infty \text{ manifolds with boundary }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the } C^\infty \text{ covering maps }\}\)
\(\{(U_{m_{2, j}} \subseteq M_2, \phi_{m_{2, j}}, \rho_{m_{2, j}}) \vert j \in J\}\): \(\in \{\text{ the } C^\infty \text{ partitions of unity subordinate to charts open cover }\}\), such that \(U_{m_{2, j}}\) is evenly-covered
\(\{(f^{-1} (U_{m_{2, j}})_l \subseteq M_1, (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \phi_{m_{2, j}}, \rho_{f^{-1} (m_{2, j})_l}) \vert j \in J, l \in L\}\), where \(L\) is the index set for sheets; \(\rho_{f^{-1} (m_{2, j})_l}: M_1 \to \mathbb{R}\) is defined as this: over \(f^{-1} (U_{m_{2, j}})_l\), \(= (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \rho_{m_{2, j}}\); over \(M_1 \setminus f^{-1} (U_{m_{2, j}})_l\), \(= 0\)
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Statements:
\(\{(f^{-1} (U_{m_{2, j}})_l \subseteq M_1, (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \phi_{m_{2, j}}, \rho_{f^{-1} (m_{2, j})_l}) \vert j \in J, l \in L\} \in \{\text{ the } C^\infty \text{ partitions of unity subordinate to the charts open cover }\}\)
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2: Note
Such a \(\{(U_{m_{2, j}} \subseteq M_2, \phi_{m_{2, j}}, \rho_{m_{2, j}}) \vert j \in J\}\) inevitably exists, as will be seen in Proof.
3: Proof
Whole Strategy: Step 1: for each \(m_2 \in M_2\), take an evenly-covered open neighborhood of \(m_2\), \(U'_{m_2}\); Step 2: take a chart around \(m_2\), \((U_{m_2} \subseteq M_2, \phi_{m_2})\), such that \(U_{m_2} \subseteq U'_{m_2}\), and possibly remove some charts to still cover \(M_2\) to have \(\{(U_{m_{2, j}} \subseteq M_2, \phi_{m_{2, j}})\}\), and see that \(\{(f^{-1} (U_{m_{2, j}})_l \subseteq M_1, (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \phi_{m_{2, j}})\}\) is a charts cover of \(M_1\); Step 3: take any \(C^\infty\) partition of unity subordinate to the charts cover of \(M_2\), \(\{\rho_{m_{2, j}}\}\), and take \(\{\rho_{f^{-1} (m_{2, j})_l}\}\), and see that it is a \(C^\infty\) partition of unity subordinate to the charts cover of \(M_1\).
Step 1:
For each \(m_2 \in M_2\), let us take an evenly-covered open neighborhood of \(m_2\), \(U'_{m_2} \subseteq M_2\).
Step 2:
Let us take a connected chart around \(m_2\), \((U_{m_2} \subseteq M_2, \phi_{m_2})\), such that \(U_{m_2} \subseteq U'_{m_2}\), which is possible, because for example, it can be taken as an \(r\)-open-ball chart around point on \(C^\infty\) manifold with boundary or an \(r\)-open-half-ball chart around point on \(C^\infty\) manifold with boundary.
The set of those charts covers \(M_2\).
The set can be used as it is, but if you want, some charts can be removed to still covers \(M_2\), and let the set be denoted as \(\{(U_{m_{2, j}} \subseteq M_2, \phi_{m_{2, j}}) \vert j \in J\}\).
Obviously, \(U_{m_{2, j}}\) is evenly-covered and \(\{f^{-1} (U_{m_{2, j}})_l \vert l \in L\}\) is the set of the sheets of \(U_{m_{2, j}}\), where \(L\) is the same for all the \(U_{m_{2, j}}\) s, by the proposition that for any covering map, the cardinalities of the sheets are the same.
\(f^{-1} (U_{m_{2, j}})_l\) is an open neighborhood of \(f^{-1} (m_2)_l\) and \(f \vert_{f^{-1} (U_{m_{2, j}})_l}: f^{-1} (U_{m_{2, j}})_l \to U_{m_{2, j}}\) is a diffeomorphism.
Let us take \(\{f^{-1} (U_{m_{2, j}})_l \vert j \in J, l \in L\}\).
It covers \(M_1\), because for each \(m_1 \in M_1\), \(f (m_1) \in U_{m_{2, j}}\) for a \(j \in J\), and \(m_1 \in f^{-1} (U_{m_{2, j}})_l\) for an \(l \in L\).
\((f^{-1} (U_{m_{2, j}})_l \subseteq M_1, (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \phi_{m_{2, j}})\) is a chart for \(M_1\), because \((f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \phi_{m_{2, j}} = \phi_{m_{2, j}} \circ f \vert_{f^{-1} (U_{m_{2, j}})_l}\) is a diffeomorphism onto an open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary, by the proposition that for any \(C^\infty\) manifold with boundary, any map from any open subset of the manifold with boundary onto any open subset of the corresponding-dimensional Euclidean \(C^\infty\) manifold or closed upper half Euclidean \(C^\infty\) manifold with boundary is a chart map if and only if it is a diffeomorphism.
So, \(\{(f^{-1} (U_{m_{2, j}})_l \subseteq M_1, (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \phi_{m_{2, j}}) \vert j \in J, l \in L\}\) is a charts cover for \(M_1\).
Step 3:
Let us take any \(C^\infty\) partition of unity subordinate to the charts cover of \(M_2\), \(\{\rho_{m_{2, j}} \vert j \in J\}\), which is possible as a well-known fact.
For each \((j, l) \in J \times L\), let us define \(\rho_{f^{-1} (m_{2, j})_l}: M_1 \to \mathbb{R}\) as follows: over \(f^{-1} (U_{m_{2, j}})_l\), \(\rho_{f^{-1} (m_{2, j})_l} = (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \rho_{m_{2, j}}\); over \(M_1 \setminus f^{-1} (U_{m_{2, j}})_l\), \(\rho_{f^{-1} (m_{2, j})_l} = 0\).
While \(supp \rho_{m_{2, j}} \subseteq U_{m_{2, j}}\) is a closed subset of \(M_2\), \(f^{-1} (supp \rho_{m_{2, j}}) \subseteq M_1\) is a closed subset, because \(f\) is continuous.
\(f^{-1} (supp \rho_{m_{2, j}}) \cap f^{-1} (U_{m_{2, j}})_l \subseteq M_1\) is a closed subset, by the proposition that for any topological space, any closed subset, and any disjoint set of open subsets whose union contains the closed subset, the intersection of the closed subset and each open subset is closed on the space: \(f^{-1} (supp \rho_{m_{2, j}}) \subseteq f^{-1} (U_{m_{2, j}})\), while \(f^{-1} (U_{m_{2, j}})\) is the union of the set of the sheets (a disjoint set of open subsets) and \(f^{-1} (U_{m_{2, j}})_l\) is one of the sheets.
The set of the nonzero points of \(\rho_{f^{-1} (m_{2, j})_l}\) is contained in \(f^{-1} (supp \rho_{m_{2, j}}) \cap f^{-1} (U_{m_{2, j}})_l\), so, \(supp \rho_{f^{-1} (m_{2, j})_l} \subseteq f^{-1} (supp \rho_{m_{2, j}}) \cap f^{-1} (U_{m_{2, j}})_l \subseteq f^{-1} (U_{m_{2, j}})_l\).
\(\rho_{f^{-1} (m_{2, j})_l}\) is \(C^\infty\), because for each \(p \in M_1\), \(p \in f^{-1} (U_{m_{2, j}})_l\) or \(p \in M_1 \setminus (f^{-1} (supp \rho_{m_{2, j}}) \cap f^{-1} (U_{m_{2, j}})_l)\) (may be in the both), and when \(p \in f^{-1} (U_{m_{2, j}})_l\), \(\rho_{f^{-1} (m_{2, j})_l} \vert_{f^{-1} (U_{m_{2, j}})_l} = \rho_{m_{2, j}} \circ f \vert_{f^{-1} (U_{m_{2, j}})_l}\) is \(C^\infty\) at \(p\), while when \(p \in M_1 \setminus (f^{-1} (supp \rho_{m_{2, j}}) \cap f^{-1} (U_{m_{2, j}})_l)\), \(\rho_{f^{-1} (m_{2, j})_l} = 0\) is \(C^\infty\) at \(p\): \(M_1 \setminus (f^{-1} (supp \rho_{m_{2, j}}) \cap f^{-1} (U_{m_{2, j}})_l)\) is an open subset of \(M_1\).
Let us take \(\{\rho_{f^{-1} (m_{2, j})_l} \vert j \in J, l \in L\}\).
Let us see that \(\{\rho_{f^{-1} (m_{2, j})_l} \vert j \in J, l \in L\}\) is a \(C^\infty\) partition of unity subordinate to \(\{(f^{-1} (U_{m_{2, j}})_l \subseteq M_1, (f \vert_{f^{-1} (U_{m_{2, j}})_l})^* \phi_{m_{2, j}}) \vert j \in J, l \in L\}\).
1) \(\forall j \in J, \forall l \in L, \forall p \in M_1 (0 \le \rho_{f^{-1} (m_{2, j})_l} (p) \le 1)\) holds, because it holds on \(f^{-1} (U_{m_{2, j}})_l\) and \(M_1 \setminus f^{-1} (U_{m_{2, j}})_l\).
2) \(\forall j \in J, \forall l \in L (supp \rho_{f^{-1} (m_{2, j})_l} \subseteq f^{-1} (U_{m_{2, j}})_l)\) has been seen above.
Let us see that 3) \(\{supp \rho_{f^{-1} (m_{2, j})_l} \vert j \in J, l \in L\} \in \{\text{ the locally finite sets of subsets of } M_1\}\).
Let \(p \in M_1\) be any.
There is an open neighborhood of \(f (p)\), \(W_{f (p)} \subseteq M_2\), that intersects only \(\{supp \rho_{m_{2, j}} \vert j \in J'\}\) where \(J' \subseteq J\) is a finite subset.
If \(f (p) \notin supp \rho_{m_{2, j}}\) for each \(j \in J'' \subset J'\), \(W_{f (p)}\) can be made smaller such that \(W_{f (p)}\) does not intersect any of \(\{supp \rho_{m_{2, j}} \vert j \in J''\}\), because as \(M_2 \setminus supp \rho_{m_{2, j}}\) is open, there is an open neighborhood of \(f (p)\), \(W'_{f (p), j} \subseteq M_2\), such that \(W'_{f (p), j} \subseteq M_2 \setminus supp \rho_{m_{2, j}}\) (so, \(W'_{f (p), j} \cap supp \rho_{m_{2, j}} = \emptyset\)), and \(W_{f (p)} \cap \cap_{j \in J''} W'_{f (p), j}\) is an open neighborhood of \(f (p)\) that intersects only \(\{supp \rho_{m_{2, j}} \vert j \in J' \setminus J''\}\). So, let us suppose that \(f (p) \in supp \rho_{m_{2, j}}\) for each \(j \in J'\).
Furthermore, \(W_{f (p)}\) can be made smaller such that it is contained in \(\cap_{j \in J'} U_{m_{2, j}}\).
Furthermore, \(W_{f (p)}\) can be made smaller such that it is evenly-covered: while there is an evenly-covered open neighborhood of \(f (p)\), the intersection of the evenly-covered open neighborhood of \(f (p)\) and \(W_{f (p)}\) is an open neighborhood of \(f (p)\), so, let us take a connected open neighborhood of \(f (p)\) contained in the intersection, which is possible because any \(C^\infty\) manifold with boundary is locally connected, and let us call the connected open neighborhood of \(f (p)\) "\(W_{f (p)}\)" again.
So, after all, \(W_{f (p)}\) is an evenly-covered open neighborhood of \(f (p)\) that intersects only \(\{supp \rho_{m_{2, j}} \vert j \in J'\}\) and is contained in \(\cap_{j \in J'} U_{m_{2, j}}\).
\(p \in f^{-1} (W_{f (p)})_l \subseteq M_1\) for an \(l \in L\), while \(f^{-1} (W_{f (p)})_l\) is an open neighborhood of \(p\).
Let us suppose that \(f^{-1} (W_{f (p)})_l \cap supp \rho_{f^{-1} (m_{2, j})_{l'}} \neq \emptyset\) for a \(j \in J\) and an \(l' \in L\).
While there is a \(p' \in f^{-1} (W_{f (p)})_l \cap supp \rho_{f^{-1} (m_{2, j})_{l'}} \subseteq f^{-1} (W_{f (p)})_l \cap f^{-1} (supp \rho_{m_{2, j}})\) (which has been seen above), \(f (p') \in W_{f (p)} \cap supp \rho_{m_{2, j}}\), which means that \(j \in J'\).
As \(W_{f (p)} \subseteq U_{m_{2, j}}\), \(f^{-1} (W_{f (p)})_l \subseteq f^{-1} (U_{m_{2, j}}) = \cup_{l' \in L} f^{-1} (U_{m_{2, j}})_{l'}\), so, \(f^{-1} (W_{f (p)})_l = f^{-1} (W_{f (p)})_l \cap \cup_{l' \in L} f^{-1} (U_{m_{2, j}})_{l'} = \cup_{l' \in L} (f^{-1} (W_{f (p)})_l \cap f^{-1} (U_{m_{2, j}})_{l'})\), by the proposition that for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset.
But each \(f^{-1} (W_{f (p)})_l \cap f^{-1} (U_{m_{2, j}})_{l'}\) is an open subset of \(f^{-1} (W_{f (p)})_l\) and \(\{f^{-1} (W_{f (p)})_l \cap f^{-1} (U_{m_{2, j}})_{l'} \vert l' \in L\}\) is disjoint, so, if there were more than \(1\) nonempty element in \(\{f^{-1} (W_{f (p)})_l \cap f^{-1} (U_{m_{2, j}})_{l'} \vert l' \in L\}\), \(f^{-1} (W_{f (p)})_l\) would not be connected, a contradiction, so, \(f^{-1} (W_{f (p)})_l \cap f^{-1} (U_{m_{2, j}})_{l'} = \emptyset\) except for only \(1\) \(l' \in L\), but as \(supp \rho_{f^{-1} (m_{2, j})_{l'}} \subseteq f^{-1} (U_{m_{2, j}})_{l'}\), \(f^{-1} (W_{f (p)})_l \cap supp \rho_{f^{-1} (m_{2, j})_{l'}} = \emptyset\) except for that only \(1\) \(l'\).
So, \(f^{-1} (W_{f (p)})_l\) intersects only some finite \(supp \rho_{f^{-1} (m_{2, j})_{l'}}\) s: \(j \in J'\) for the finite \(J'\) and only \(1\) \(l'\) for each \(j\).
Let us see that 4) \(\forall p \in M_1 (\sum_{j \in J, l \in L} \rho_{f^{-1} (m_{2, j})_l} (p) = 1)\).
Let \(J' := \{j \in J \vert p \in f^{-1} (U_{m_{2, j}})\}\).
For each \(j \in J'\), \(p \in f^{-1} (U_{m_{2, j}})_l\) for an only \(1\) \(l \in L\), and as \(l\) depends on \(j\), let us denote that \(p \in f^{-1} (U_{m_{2, j}})_{l (j)}\).
\(\sum_{j \in J, l \in L} \rho_{f^{-1} (m_{2, j})_l} (p) = \sum_{j \in J'} \rho_{f^{-1} (m_{2, j})_{l (j)}} (p) = \sum_{j \in J'} \rho_{m_{2, j}} \circ f (p) = \sum_{j \in J'} \rho_{m_{2, j}} (f (p))\).
\(p \in f^{-1} (U_{m_{2, j}})\) is nothing but \(f (p) \in U_{m_{2, j}}\), so, \(J' = \{j \in J \vert f (p) \in U_{m_{2, j}}\}\).
So, \(\sum_{j \in J'} \rho_{m_{2, j}} (f (p)) = \sum_{j \in J} \rho_{m_{2, j}} (f (p)) = 1\).
So, \(\sum_{j \in J, l \in L} \rho_{f^{-1} (m_{2, j})_l} (p) = 1\) after all.
