Number of Differentiable Structures on a Smooth Manifold

Question

On John Lee's book, Introduction to Smooth Manifolds, I stumbled upon the next problem (problem 1.6):

Let $M$ be a nonempty topological manifold of dimension $n \geq 1$. If $M$ has a smooth structure, show that it has uncountably many distinct ones.

The trick in this exercise was to use the function $F_s(x) = |x|^{s-1}x$, where $s \in \mathbb{R}$ and $s>0$. This function defines an homeomorphism from $\mathbb{B}^n$ to itself, and is a diffeomorphism iff $s=1$.

Now, reading Loring W. Tu's book, An Introduction to Manifolds, he writes:

"It is known that in dimension $< 4$ every topological manifold has a unique differentiable structure and in dimension $>4$ every compact topological manifold has a finite number of differentiable structures. [...]"

Can someone help me explain how this last "known fact" and problem 1.6 in Lee's book don't contradict each other?

Thanks in advance

Answer 1

The distinction to be made is that a differentiable structure is a choice of maximal smooth atlas $\mathcal A$, but two different choices $\mathcal A$ and $\mathcal A'$ can lead to isomorphic smooth structures. As an example, the canonical smooth structure $\mathcal A$ on $\mathbb R$ that contains the smooth function ${\rm id}:\mathbb R\longrightarrow \mathbb R$ is isomorphic to the smooth structure $\mathcal A'$ that contains the smooth function $x\mapsto x^3$, although $\mathcal A'\neq \mathcal A$. Thus, although a manifold admits uncountably many different smooth structures, it may have finitely many isomorphism classes of such structures.

Answer 2

In the second statement, "unique" means unique up to diffeomorphism.

If you have a manifold $M$ with a smooth structure $A$ and a homeomorphism $\varphi :M \rightarrow M$, which is not a diffeomorphism if we consider it as a map between the smooth manifolds $(M,A) \rightarrow (M,A)$, then we can define a distinct smooth structure, say A', on $M$ by composing the coordinate charts of $M$ with $\varphi$.

Now consider $\varphi: (M,A') \rightarrow (M,A)$. Which this respect to these smooth structures, $\varphi$ will be a diffeomorphism. So while you have a distinct smooth structure, it is not really that different.

The (quite difficult) question how many smooth structures on a given topological manifold exist up to diffeomorphism. This is what Tu talks about.