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I am looking for a reference of the following result, that I read a while ago in a book but I cannot remember where:

Any (compact?) $n$-manifold, $n \geq 5$, has a finite number of smooth structures. In other words, if $M$ is a topological manifold, there exists a finite number of homeomorphic, non-diffeomorphic smooth manifolds $M_{1}, \ldots, M_{k}$ such that $M$ is homeomorphic to all of them and any smooth structure on $M$ is equivalent to the one of some $M_i$.

Maybe $k$ coincides with the number of smooth structures on $S^n$, but I cannot remember the whole statement.

Minkowski
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2 Answers2

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The statement you refer to is a theorem of Kirby and Siebenmann; see this answer for more information.

As for your last statement, the number $k$ is not always equal to the number of smooth structures on $S^n$. For $n \geq 5$, the number of isomorphism classes of smooth structures on $T^n$ is in one-to-one correspondence with $H^3(T^n; \mathbb{Z}_2)$; see this MathOverflow answer. Therefore $T^n$ has $2^{\binom{n}{3}}$ smooth structures, while the number of smooth structures on $S^n$ is much more complicated, as discovered by Kervaire and Milnor; see here and here.

For example, $T^5$ has $2^{10} = 1024$ different smooth structures, while $S^5$ has only one.

Michael Albanese
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There are not really as many structures on $T^n$ as that: typically one mods out $H^3(T^n;\mathbb{Z}/2)$ by the action of $GL(n,\mathbb{Z}/2)$. This is discussed by Wall in chapter 15A of Surgery on Compact Manifolds. In fact up to diffeomorphism there are 3 smooth structures on $T^5$. It's important to also credit Hsiang-Shaneson here.

Nonetheless, $T^n$ does have multiple smooth structures up to diffeomorphism, for all $n \geq 5$, and typically much more than $S^n$ does, so the general idea behind the answer is correct.

The answer also missed the action of the group of homotopy spheres $\Theta_n$ on the set of diffeomorphisms classes of smooth structures on $T^n$. This is a nontrivial action when $\Theta_n$ is nontrivial, which does not change the corresponding element of $H^3(T^n;\mathbb{Z}/2)$. So the exotic spheres also give rise to exotic tori.

Hsiang, Wu-chung; Shaneson, Julius L. "Fake tori." In: 1970 Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) pp. 18–51 Markham, Chicago, Ill.

Dave Davidson
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