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This question was part of exercises in my manifold course which I am taking online.

Question : There exists two different atlases on $S^{7} \subseteq \mathbb{R}^8$ that are not diffeomorphic. ( exotic sphere)

Attempt: I have to show that two atlases exists by constructing them. This seemed to be a really hard job.

On the other hand , I thought if I could assume that there doesn't exists two atlases that are not diffeomorphic ie each atlases is diffeomorphic to other atlas.

So, { $(U_{\alpha},\phi_{\alpha})$} , {$ ( V_{\beta} , \psi_{\beta})$} are diffeomorphic. But, I am really sorry that I am not able to move forward from here.

I have been following Frank Warner's Introduction to differntial manifolds.

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    $\begingroup$ More strongly than just atlases, there exist two smooth structures (aka maximal atlases) on $S^7$ which are homeomorphic but not diffeomorphic. There are number of them in fact, and Wikipedia should have information on them... Look up "exotic spheres" $\endgroup$ Sep 28, 2022 at 18:15
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    $\begingroup$ This is a really not trivial Theorem, not an exercise! $\endgroup$
    – Didier
    Sep 28, 2022 at 19:50
  • $\begingroup$ Google search with keywords "exotic sphere atlas" and select the "image" option: you will see inspiring things... $\endgroup$
    – Jean Marie
    Sep 28, 2022 at 21:59
  • $\begingroup$ @Didier Thank you very much for your advice! So, can you please tell if this question/ theorem can be given in 1st course on Manifolds and that too in the 1st part of the course? $\endgroup$
    – user775699
    Sep 29, 2022 at 10:21
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    $\begingroup$ @Avenger I am sure that no. In my opinion, this is too advanced: Milnor's construction requires knowledge on fibre bundles (here, $S^3$-bundles over $S^4$), Morse theory, Characteristic classes, and some basic algebraic topology (for instance, how to compute $\pi_1(SO(4))$). You can try to read this explanatory article $\endgroup$
    – Didier
    Sep 29, 2022 at 10:31

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Just to be clear, this is in no sense an "exercise" unless you were specifically fed a bunch of results leading up to it. Exotic $7$-spheres were first constructed by Milnor in his 1956 Annals paper On manifolds homeomorphic to the $7$-sphere, and as far as I know this was the first construction of an exotic smooth structure to appear in the mathematical literature. This was not at all an easy or trivial result. I have no idea what whoever assigned this as an "exercise" had in mind.

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There's $28$ $S^7$'s and they form an abelian group. John Milnor discovered them, and probably won a Field's medal for it.

They're homeomorphic but not diffeomorphic.

In any dimension other than $4$ the differentiable structures on $S^n$ form a finite abelian group. (They always form an abelian monoid.)

Here's one of his first constructions.

Now I am paraphrasing (because I find this interesting). Apparently he proved it wasn't diffeomorphic to the standard $S^7$ by showing it isn't the boundary of any smooth $8$-manifold with vanishing $4$-th Betti number. Alternatively it has no orientation reversing diffeomorphism.

He used Morse theory to prove they're homeomorphic. How many, or if, exotic $4$-spheres exist is an unsolved problem.

By the generalized Poincare conjecture all homotopy $n$-spheres are homeomorphic to $S^n$. (John Stallings, whom I had the pleasure of studying under as an undergraduate, did dimension $6$.)

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