Twisted Spheres
Given an (orientation-preserving) diffeomorphism f: Sn−1→Sn−1, gluing the boundaries of two copies of the standard disk Dn together by 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. (Milnor 1959b) Setting to be the group of twisted n-spheres (under connect sum), one obtains the exact sequence
For n > 4, every exotic sphere is diffeomorphic to a twisted sphere, a result proven by Stephen Smale. (In contrast, in the piecewise linear setting the left-most map is onto via radial extension: there are no piecewise-linear-twisted spheres.) The group Γn of twisted spheres is always isomorphic to the group Θn. The notations are different because it was not known at first that they were the same for n=3 or 4; for example, the case n=3 is equivalent to the Poincaré conjecture.
In 1970 Jean Cerf proved the pseudoisotopy theorem which implies that is the trivial group provided, so provided .
Read more about this topic: Exotic Sphere
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