Every smooth atlas \( \mathcal{A} \) on a manifold \( M \) is contained in a unique maximal smooth atlas , called the smooth structure determined by \( \mathcal{A} \) [1, Proposition 1.17].
- We use this statement to show that a manifold \(M\) has a smooth structure. Then it is sufficient to find some smooth atlas on \(M\), and by this proposition there exists a smooth structure on \(M\).
Let \( \bar{\mathcal{A}} \) denote the set of all charts that are smoothly compatible with every chart in \( \mathcal{A} \).
Then \( \bar{\mathcal{A}} \) is a smooth atlas. To see this, we need to prove that \( (U, \varphi), (V, \psi) \in \bar{\mathcal{A}} \) are smoothly compatible. For every point \( p \in U \cap V \), we find a chart in \( \mathcal{A} \); we can use it to prove that \( \psi \circ \varphi^{-1} \) is a diffeomorphism in a neighbourhood of \( \varphi(p) \).
The set \( \bar{\mathcal{A}} \) is maximal since if a chart in \(\bar{\mathcal{A}}\) is smoothly compatible with every chart in \( \bar{\mathcal{A}} \), it must in particular be compatible with every chart in \( \mathcal{A} \), so it is in \( \bar{\mathcal{A}} \).
If \( \mathcal{B} \) is another maximal atlas containing \( \mathcal{A} \), each of its charts is compatible with every chart in \( \mathcal{A} \), and thus \( \mathcal{B} \subseteq \bar{\mathcal{A}} \). By maximality of \( \mathcal{B} \), we have \( \mathcal{B} = \bar{\mathcal{A}} \).
References Link to heading
- J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.