Editing Exotic sphere (section)

==Explicit examples of exotic spheres==

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|quote=  When I came upon such an example in the mid-50s, I was very puzzled and didn't know what to make of it. At first, I thought I'd found a counterexample to the generalized Poincaré conjecture in dimension seven. But careful study showed that the manifold really was homeomorphic to <math>S^7</math>. Thus, there exists a differentiable structure on <math>S^7</math> not diffeomorphic to the standard one.
|source={{harvs|txt|first=John|last=Milnor|year=2009|loc=p.12}}
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=== Milnor's construction ===
{{Main|Milnor's sphere}}
One of the first examples of an exotic sphere found by {{harvtxt|Milnor|1956|loc=section 3}} was the following. Let <math id="en.wikipedia.org/wiki/Ball_(Mathemathics)">B^4</math> be the unit ball in <math>\R^4</math>, and let <math>S^3</math> be its [[Boundary (topology)|boundary]]—a 3-sphere which we identify with the group of unit [[quaternion]]s. Now take two copies of <math>B^4 \times S^3</math>, each with boundary <math>S^3 \times S^3</math>, and glue them together by identifying <math>(a,b)</math> in the first boundary with <math>(a,a^2ba^{-1})</math> in the second boundary. The resulting manifold has a natural smooth structure and is homeomorphic to <math>S^7</math>, but is not diffeomorphic to <math>S^7</math>. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a [[Morse function]] with just two [[critical point (mathematics)|critical points]], both non-degenerate, which implies that it is topologically a sphere.

=== Brieskorn spheres ===
{{Main|Brieskorn manifold}}

As shown by {{harvs|txt=yes|first=Egbert |last=Brieskorn|author-link=Egbert Brieskorn|year=1966|year2=1966b}} (see also {{harv|Hirzebruch|Mayer|1968}}) the intersection of the [[complex manifold]] of points in <math>\Complex^5</math> satisfying
:<math>a^2 + b^2 + c^2 + d^3 + e^{6k-1} = 0\ </math>
with a small sphere around the origin for <math>k = 1, 2, \ldots, 28</math> gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called [[Brieskorn sphere]]s.