definition of \(C^\infty\) covering map
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of covering map.
- The reader knows a definition of \(C^k\) map between arbitrary subsets of \(C^\infty\) manifolds with boundary, where \(k\) includes \(\infty\).
- The reader knows a definition of diffeomorphism between arbitrary subsets of \(C^\infty\) manifolds with boundary.
Target Context
- The reader will have a definition of \(C^\infty\) covering map.
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:
\( M_1\): \(\in \{\text{ the connected and locally path-connected } C^\infty \text{ manifolds with boundary }\}\)
\( M_2\): \(\in \{\text{ the connected and locally path-connected } C^\infty \text{ manifolds with boundary }\}\)
\(*\pi\): \(: M_1 \to M_2\), \(\in \{\text{ the surjections }\} \cap \{\text{ the } C^\infty { maps }\}\)
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Conditions:
\(\forall m \in M_2 (\exists U_m \subseteq M_2 \in \{\text{ the open neighborhoods of } m\} \text{ such that } U_m \text{ is } C^\infty \text{ evenly-covered by } \pi)\), where "\(C^\infty\) evenly-covered by \(\pi\)" means that for each connected component of \(\pi^{-1} (U_m)\), \(\pi^{-1} (U_m)_j\), where \(j \in J\) where \(J\) is a possibly uncountable index set, \(\pi \vert_{\pi^{-1} (U_m)_j}: \pi^{-1} (U_m)_j \to U_m\) is a diffeomorphism
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\(M_2\) is called "base of covering".
\(M_1\) is called "covering space of \(M_2\)".
Each \(\pi^{-1} (U_m)_j\) is called "sheet of covering over \(U_m\)".
