Exotic Sphere - Twisted Spheres

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 Γ_n of twisted spheres is always isomorphic to the group Θ_n. 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 .