Confused about the possibility of different differentiable structures.

Solution 1:

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

Solution 2:

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).