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In the book Foundations of Differentiable Manifolds and Lie Groups (by Warner) the author defines a differentiable manifold to be a pair $(M, F)$ where $F$ is a maximal atlas. Usually in the literature, I have seen $F$ to be any atlas.

Why would an author require the atlas to be maximal? What will be gained by it?

  • If you would like to look it up in the book: it is on page 5 and 6. (bottom of 5 and middle of 6) –  Oct 12 '14 at 11:49

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One shouldn't define a smooth (differentiable) manifold to be a pair $(M, F)$ where $F$ is some atlas---if we used this definition, then given any $(M, F)$ and an atlas $F'$ compatible with but distinct from $F$, $(M, F)$ and $(M, F')$ would be different smooth manifolds. One way to rectify this situation is to define a smooth manifold to be an equivalence class of pairs $(M, F)$, where $(M, F) \sim (M', F')$ iff (a) $M = M'$ and (b) $F$ and $F'$ are compatible.

On the other hand, if $F$ and $F'$ are compatible, then by definition they determine the same maximal smooth atlas, so if we simply demand that the atlas $F$ in a pair $(M, F)$ be maximal, then we can avoid involving equivalence classes as above, because by construction maximality already entails the appropriate equivalence.

Put another way, picking an atlas of a smooth manifold involves some choice, but a maximal smooth atlas is a built-in feature of a smooth manifold, and in particular involves no choice.

Travis Willse
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  • Thank you for this great answer. One question: when you write choice, do you mean axiom of choice? –  Oct 13 '14 at 00:15
  • You're welcome, I'm glad you found it useful. By choice I just meant that by construction a smooth manifold has a single maximal atlas, and if you like, this maximal atlas is the smooth structure on the underlying topological manifold. On the other hand, there are many (many) atlases that define the same smooth structure, and when we specify any of them, we are making a choice, and hence prescribing more information than is contained in the smooth structure itself, namely, a declaration of which compatible charts are and are not in our atlas. – Travis Willse Oct 13 '14 at 05:00
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    I'm not an expert in AC, but I suspect you can avoid invoking it when defining a smooth structure, but you may not be able to avoid it when selecting an atlas as a subset of a maximal atlas that contains no finite atlases. I'd be grateful if anyone could illuminate this point for me. – Travis Willse Oct 13 '14 at 05:02