1062: C∞ Covering Map

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definition of $C^{\mathrm{\infty }}$ covering map

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description

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Starting Context

• The reader knows a definition of covering map.
• The reader knows a definition of $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.
• The reader knows a definition of diffeomorphism between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary.

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Target Context

• The reader will have a definition of $C^{\mathrm{\infty }}$ covering map.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M_1$: $\in \{\text{ the connected and locally path-connected }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the connected and locally path-connected }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\ast \pi$: $:M_1\to M_2$, $\in \{\text{ the surjections }\}\cap \{\text{ the }{C}^{\mathrm{\infty }}maps\}$
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Conditions:
$\mathrm{\forall }m\in M_2(\mathrm{\exists }{U}_m\subseteq M_2\in \{\text{ the open neighborhoods of }m\}\text{ such that }{U}_m\text{ is }{C}^{\mathrm{\infty }}\text{ evenly-covered by }\pi )$, where "$C^{\mathrm{\infty }}$ 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, $\pi {|}_{{\pi }^{-1}(U_m{)}_j}:{\pi }^{-1}(U_m{)}_j\to U_m$ 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 $U_m$".

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References

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