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?
Best Answer
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 $f(x) = \sqrt[3]{x}$.