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In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.

He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.

My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept?

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    $\begingroup$ "Maximal" usually means "cannot be further extended", but does not imply uniqueness. For instance a maximal ideal is one any extension of which is either itself or the whole ring. A ring may have many distinct maximal ideals. $\endgroup$ Jun 30 '14 at 16:04
  • $\begingroup$ Thank you. I think I understand this concept then. But my question still remains in this case. Are all coordinate maps compatible with each other as a result of being diffeomorphisms? Would this not imply that there could only be one maximal atlas on a manifold by our construction regardless of dimension? $\endgroup$
    – Memeozuki
    Jun 30 '14 at 16:12
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    $\begingroup$ All coordinate maps within the same atlas are certainly compatible with each other. As my answer shows, there will always exist coordinate maps which are incompatible with a given atlas. Such coordinate maps may form the foundation of an entirely different maximal atlas for an entirely different differentiable structure. $\endgroup$
    – Lee Mosher
    Jun 30 '14 at 18:31
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You cite from Tu's book a statement that "for any chart on a manifold, the coordinate map is a diffeomorphism onto its image", but if you check that statement carefully I'm sure that it applies only to charts within the given maximal atlas. It is certainly possible to have incompatible coordinate charts that are not in a given maximal atlas, e.g. here are two incompatible coordinate charts on $\mathbb{R}$: $f(x)=x$; and $f(x) = \sqrt[3]{x}$.

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  • $\begingroup$ Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($\mathbb{R}$), right?. $\endgroup$ Nov 19 '18 at 10:30
  • $\begingroup$ That's correct. $\endgroup$
    – Lee Mosher
    Nov 19 '18 at 14:04
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This is to correct the statement that you quote from Tu's book:

"Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures"

What Tu actually writes (a bit sloppily) on page 56 is:

It is known that in dimensions < 4 every topological manifold has a unique differentiable structure and in dimensions > 4 every compact topological manifold has a finite number of differentiable structures. Dimension 4 is a mystery.

First of all, Tu means that the given compact topological manifold $X$ of dimension $\dim(X)>4$ has only finitely many differentiable structures up to diffeomorphism, i.e. there is a finite (possibly empty) set $S$ smooth atlases on $X$ such that for every smooth atlas ${\mathcal A}$ on $X$, the smooth manifold $(X, {\mathcal A})$ is diffeomorphic to $(X, {\mathcal B})$ for some ${\mathcal B}\in S$. Note that one has to impose the strict inequality $\dim(X)>4$ since there exist 4-dimensional compact manifolds which admit infinitely many pairwise non-diffeomorphic smooth structures. (See my answer here.) This result goes back to late 1980s, so the situation in dimension 4 was not a complete mystery when Tu wrote his book (although, much indeed remains unknown).

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