Editing Exotic sphere (section)

=== 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.