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I am working through Loring Tu's An Introduction to Manifolds (Second Edition). On page no. 56, he writes:

One of the most surprising achievements in topology was John Milnor’s discovery [27] in 1956 of exotic $7$-spheres, smooth manifolds homeomorphic but not diffeomorphic to the standard $7$-sphere. In 1963, Michel Kervaire and John Milnor [24] determined that there are exactly $28$ nondiffeomorphic differentiable structures on $S^7$.

I think that the first sentence states what exotic $7$-spheres are. They are smooth manifolds (i.e., topological manifolds with own maximal atlases) that are homeomorphic but not diffeomorphic to the standard $7$-sphere, where the standard $7$-sphere is the topological manifold $S^7$ with a maximal atlas.

My problem is with the second sentence. It says that there are exactly $28$ nondiffeomorphic differentiable structures on $S^7$. Nondiffeomorphic to what?

Also, a differentiable structure or maximal atlas is just a set of charts (which of course meet the requirements in the definition of a maximal atlas.) We know that a diffeomorphism of manifolds is a bijective $C^{\infty}$ map $F: N \to M$ whose inverse $F^{-1}$ is also $C^{\infty}$, where $M$ and $N$ are smooth manifolds. What is the definition of the diffeomorphism of two atlases, so that I understand what nondiffeomorphic differentiable structures mean?

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3 Answers 3

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Perhaps a better way to write this would be "pairwise non-diffeomorphic". This means that no two manifolds of the 28 are diffeomorphic to one another. The definition of diffeomorphism is the one you know.

Thus, there are 28 smooth manifolds, $M_1,...,M_{28}$, all homeomorphic to the 7-sphere, such that:

  1. If $i\neq j$, then there is no diffeomorphism $F:M_i\to M_j$;
  2. If $\tilde{M}$ is a smooth manifold which is homeomorphic to the 7-sphere, there is some $i\in \{1,..,28\}$ such that $\tilde{M}$ is diffeomorphic to $M_i$.

I hope it's clearer now.

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The concept of smoothness depends on the atlas itself. So, for this question, it might pay off to write a smooth manifold as a pair $(M,\mathfrak{A})$, where $M$ is a topological manifold and $\mathfrak{A}$ is a maximal smooth atlas for $M$. Then a diffeomorphism between two atlases $\mathfrak{A}_1$ and $\mathfrak{A}_2$ on the same manifold is just a diffeomorphism $(M,\mathfrak{A}_1) \to (M,\mathfrak{A}_2)$.

Regarding the situation about $\Bbb S^7$, it means that you have $28$ maximal atlases $\{ \mathfrak{A}_1 = \mathfrak{A}_{\rm std}, \mathfrak{A}_2,\ldots, \mathfrak{A}_{28} \}$ such that if $i,j \in \{1,\ldots, 28\}$ are distinct, then $(\Bbb S^7,\mathfrak{A}_i)$ and $(\Bbb S^7, \mathfrak{A}_j)$ are not diffeomorphic, and if $\mathfrak{A}$ is any maximal atlas for $\Bbb S^7$, there is $i \in \{1,\ldots, 28\}$ such that $(\Bbb S^7,\mathfrak{A})$ is diffeomorphic to $(\Bbb S^7,\mathfrak{A}_i)$.


Edited to add reference: see pp. 9--12 in Differential Geometric Structures by Walter Poor.

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Actually, there are 15 smooth structures on the 7-sphere up to diffeomorphism. Provided one allows orientation-reserving diffeomorphisms. The group of isotopy classes of smooth structures on the 7-sphere is isomorphic to $\mathbb{Z}/28$, as famously proven by Kervaire and Milnor. A reflection acts by multiplication by $-1$, and there are 15 orbits.

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