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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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2 Answers 2

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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, 2016 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, 2017 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 ♦
    Feb 7, 2017 at 17:42
  • $\begingroup$ It's maybe worth adding that although it may not be immediate that A or A' are maximal, their corresponding maximal smooth atlases can not be equal because the two coordinate functions chosen are not smoothly compatible (and thus must live in different maximal atlases by definition). $\endgroup$
    – D. Zack Garza
    Jun 22, 2020 at 23:18
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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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