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I'm reading "An Introduction to Manifolds" by Tu. In this book, the definition of a topological manifold is a Hausdorff, second countable locally Euclidean space and the definition of a smooth manifold is a topological manifold with a maximal (pairwise $C^\infty$-compatible) atlas.

I know that given a topological manifold $M$ and an atlas $\mathfrak{A}$ on $M$, there is a unique maximal atlas on $M$ that contains $\mathfrak{A}$. My question is that is it possible to have two distinct maximal atlases on the topological manifold $M$ (that is,they are not required to contain a given atlas on $M$)?

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  • $\begingroup$ If I understand correctly, then Proposition 5.10 of Tu may be understood in the following way as illustrated in the answer by Alex Ortiz. If $A_1$ and $A_2$ are two atlases which are not compatible with each other, then they are contained in two different maximal atlases $\mathcal{M}_1$ and $\mathcal{M}_2$, where $\mathcal{M}_1 \neq \mathcal{M}_2$. $\endgroup$
    – rainman
    Commented Jun 10, 2020 at 16:43

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In general there is not a unique maximal atlas on the manifold $M$. Consider $\Bbb R$ and the charts $(\Bbb R,\operatorname{Id})$ and $(\Bbb R,f)$, where $f(x) = x^3$. Each of these charts cover $\Bbb R$, but it's easy to check they are not smoothly compatible, so they generate distinct maximal atlases. However, the smooth manifolds we get by equipping $\mathbb R$ with each of these maximal atlases are diffeomorphic.

Exotic spheres are a famous example of spaces having multiple smooth structures that are not diffeomorphic to one another.

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