# Standard 7-Sphere

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  in 1956 of exotic $$7$$-spheres, smooth manifolds homeomorphic but not diffeomorphic to the standard $$7$$-sphere. In 1963, Michel Kervaire and John Milnor  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?

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 $$M_i$$.
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 $$(\Bbb S^7,\mathfrak{A}_i)$$.