Professor said there are 28 $\mathfrak{C}^{k}$ structures on $\mathbb{S} ^7$ during lesson. Then I checked myself and came across with others as: 992 for $\mathbb{S} ^{11} $ or just 1 for $\mathbb{S} ^{12} $. How can we calculate the number of structures?
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3I suspect that this is too difficult a problem for a detailed answer here. The keyword to search for is exotic spheres. See for example the classification section in the wikipedia page [here](https://en.m.wikipedia.org/wiki/Exotic_sphere). – Rhys Steele Oct 29 '18 at 14:15

1Note also that there are many, many references and external links on Wiki's "exotic sphere" page (the page [differential structures](https://en.wikipedia.org/wiki/Differential_structure#Differential_structures_on_spheres_of_dimension_1_to_20) that the OP found has minimal references.) I did a quick edit on the Wiki article to make the link more explicit. – Michael Seifert Oct 29 '18 at 14:17

Is a $\mathfrak{C}^{k}$ structure the same thing as a $C^k$ structure? – David C. Ullrich Oct 29 '18 at 15:14

Yes, exactly. I think the book which I follow uses different notation. – Emre Yılmaz Oct 30 '18 at 16:04

1This is a major and very difficult question. There is no general formula, and it is not even known if all sufficiently high dimensional spheres always have exotic smooth structures. I wrote an answer about the difficult algebraic topology involved [here](https://math.stackexchange.com/a/1609522/98602). – Nov 02 '18 at 15:49