484: Maximal Atlas for Topological Manifold with Boundary

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A definition of maximal atlas for topological manifold with boundary

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Topics

About: topological manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of topological manifold with boundary.
• The reader knows a definition of chart on topological manifold with boundary.
• The reader knows a definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.

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Target Context

• The reader will have a definition of maximal atlas for topological manifold with boundary.

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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: Definition

For any topological manifold with boundary, $M$, any set of mutually $C^{\mathrm{\infty }}$ compatible charts that covers $M$ to which (the set) any possible $C^{\mathrm{\infty }}$ compatible chart has been already added, where "mutually $C^{\mathrm{\infty }}$ compatible charts" means that for 2 charts $(U_1\subseteq M,{\varphi }_1)$ and $(U_2\subseteq M,{\varphi }_2)$, ${\varphi }_2\circ {{\varphi }_1}^{-1}{|}_{{\varphi }_1(U_1\cap U_2)}:{\varphi }_1(U_1\cap U_2)\to {\varphi }_2(U_1\cap U_2)$ and ${\varphi }_1\circ {{\varphi }_2}^{-1}{|}_{{\varphi }_2(U_1\cap U_2)}:{\varphi }_2(U_1\cap U_2)\to {\varphi }_1(U_1\cap U_2)$ are $C^{\mathrm{\infty }}$ at each point (when $U_1\cap U_2=\mathrm{\varnothing }$, the charts are vacuously $C^{\mathrm{\infty }}$ compatible), where while ${\varphi }_j(U_1\cap U_2)$ is not necessarily open on ${\mathbb{R}}^d$, $C^{\mathrm{\infty }}$-ness is by the definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$

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2: Note

We could talk about "the maximal continuous atlas" as the set of mutually continuously compatible charts that covers $M$, but we do not because there is no choice for it (it is uniquely determined, because any 2 charts are inevitably continuously compatible); we especially talk about "maximal atlas" by this definition because it is a matter of choosing an atlas from some possibly multiple candidates.

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References

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