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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

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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.

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  • $\begingroup$ Gracias Pedro :) $\endgroup$ – rie Aug 27 '16 at 18:04
  • $\begingroup$ "a differentiable structure is a choice of maximal smooth atlas $A$". How is it possible to have multiple maximal smooth atlasses? If atlasses $A'$ and $A$ are both smooth atlasses, isn't their union also a smooth atlas? (thus contradicting the assumption that they are maximal smooth atlasses). $\endgroup$ – user56834 Feb 7 '17 at 12:39
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    $\begingroup$ @Programmer2134 No, the union of two smooth atlasses is not always a smooth atlas. This is true if and only if they are contained in the same maximal atlas. $\endgroup$ – Pedro Tamaroff Feb 7 '17 at 17:42
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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.

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