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Wikipedia says:

As shown by Egbert Brieskorn (1966, 1966b) (see also Hirzebruch & Mayer 1968) the intersection of the complex manifold of points in $\mathbb{C}^5$ satisfying $$a^2+b^2+c^2+d^3+e^{6k-1},$$ with a small sphere around the origin for $k = 1, 2, \dotsb, 28$ gives all $28$ possible smooth structures on the oriented $7$-sphere. Similar manifolds are called Brieskorn spheres.

Wikipedia gives three sources, but the two (linked) that may actually give the proof or expand on the above result are not in English. Are these $28$ structures the $28$ exotic spheres? Are there any known English translations or papers/books expanding on the facts above? Any insights or related articles classifying the exotic $7$-spheres would be greatly appreciated.

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    $\begingroup$ Yes, the 28 structures are the 28 exotic spheres. An english source is Milnor and Kervair's Groups of Homotopy spheres I, which appeared in the Annals. Here's a copy maths.ed.ac.uk/~aar/papers/kervmiln.pdf $\endgroup$
    – Jason DeVito - on hiatus
    Mar 16, 2013 at 1:57
  • $\begingroup$ @Jason: That looks like a full-fledged answer to me :-) $\endgroup$
    – joriki
    Mar 16, 2013 at 4:47
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    $\begingroup$ @joriki: You're right. I honestly read the question in haste and thought a much longer reply was in order. I'll turn this into an answer in a second... $\endgroup$
    – Jason DeVito - on hiatus
    Mar 16, 2013 at 13:36

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Yes, the 28 differentiable structures are the 28 exotic spheres. If you want a source in English, it's hard to beat the original:

Kervaire and Milnor, Groups of homotopy spheres: I, Ann. Math. (2) 77 (1963), 504-537

Their paper is also available for download here, which I believe is off of Andrew Ranicki's website.

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  • $\begingroup$ The original paper has a mistake in the computation of one of the $\Theta_n$s, but I don't remember which one. If you remind me, I can look it up on Monday in Milnor's collected works where he has a footnote detailing exactly which one is wrong. Or maybe I'm just misremembering. $\endgroup$
    – Jason DeVito - on hiatus
    Mar 16, 2013 at 13:49
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    $\begingroup$ Wikipedia lists the orders of the groups $\Theta_n$, and says that there was an error for $n=19$ in the original paper. $\endgroup$
    – Mike Pierce
    Nov 26, 2016 at 5:36

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