A definition of maximal atlas for topological manifold with boundary
Topics
About: topological manifold
The table of contents of this article
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 \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
Target Context
- The reader will have a definition of maximal atlas for topological manifold with boundary.
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: Definition
For any topological manifold with boundary, \(M\), any set of mutually \(C^\infty\) compatible charts that covers \(M\) to which (the set) any possible \(C^\infty\) compatible chart has been already added, where "mutually \(C^\infty\) compatible charts" means that for 2 charts \((U_1 \subseteq M, \phi_1)\) and \((U_2 \subseteq M, \phi_2)\), \(\phi_2 \circ {\phi_1}^{-1}\vert_{\phi_1 (U_1 \cap U_2)}: \phi_1 (U_1 \cap U_2) \to \phi_2 (U_1 \cap U_2)\) and \(\phi_1 \circ {\phi_2}^{-1}\vert_{\phi_2 (U_1 \cap U_2)}: \phi_2 (U_1 \cap U_2) \to \phi_1 (U_1 \cap U_2)\) are \(C^\infty\) at each point (when \(U_1 \cap U_2 = \emptyset\), the charts are vacuously \(C^\infty\) compatible), where while \(\phi_j (U_1 \cap U_2)\) is not necessarily open on \(\mathbb{R}^d\), \(C^\infty\)-ness is by the definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\)
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.
