Editing Exotic sphere (section)

==Twisted spheres==
Given an (orientation-preserving) diffeomorphism <math>f\colon S^{n-1} \to   S^{n-1}</math>, gluing the boundaries of two copies of the standard disk <math>D^n</math> together by ''f'' yields a manifold called a ''twisted sphere'' (with ''twist'' ''f''). It is homotopy equivalent to the standard ''n''-sphere because the gluing map is homotopic to the identity (being an orientation-preserving diffeomorphism, hence degree 1), but not in general diffeomorphic to the standard sphere. {{harv|Milnor|1959b}}
Setting <math>\Gamma_n</math> to be the group of twisted ''n''-spheres (under connect sum), one obtains the exact sequence
:<math>\pi_0\operatorname{Diff}^+(D^n) \to \pi_0\operatorname{Diff}^+(S^{n-1}) \to \Gamma_n \to 0.</math>

For <math>n>5</math>, every exotic ''n''-sphere is diffeomorphic to a twisted sphere, a result proven by [[Stephen Smale]] which can be seen as a consequence of the [[H-cobordism#Precise statement of the h-cobordism theorem|''h''-cobordism theorem]]. (In contrast, in the [[Piecewise linear manifold|piecewise linear]] setting the left-most map is onto via [[Alexander Trick#Radial extension|radial extension]]: every piecewise-linear-twisted sphere is standard.) The group <math>\Gamma_n</math> of twisted spheres is always isomorphic to the group <math>\Theta_n</math>. The notations are different because it was not known at first that they were the same for <math>n = 3</math> or 4; for example, the case <math>n = 3</math> is equivalent to the [[Poincaré conjecture]].

In 1970 [[Jean Cerf]] proved the [[pseudoisotopy theorem]] which implies that <math>\pi_0 \operatorname{Diff}^+(D^n)</math> is the trivial group provided <math>n \geq 6</math>, and so <math>\Gamma_n \simeq \pi_0 \operatorname{Diff}^+(S^{n-1})</math> provided <math>n \geq 6</math>.