Brieskorn spheres As shown by (see also ) the intersection of the complex manifold of points in satisfying with a small sphere around the origin for gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called Brieskorn spheres. Twisted spheres Given an (orientation-preserving) diffeomorphism , gluing the boundaries of two copies of the standard disk 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. Setting to be the group of twisted n-spheres (under connect sum), one obtains the exact sequence For , 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 theorem. (In contrast, in the piecewise linear setting the left-most map is onto via radial extension: every piecewise-linear-twisted sphere is standard.) The group of twisted spheres is always isomorphic to the group . The notations are different because it was not known at first that they were the same for or 4; for example, the case is equivalent to the Poincaré conjecture. In 1970 Jean Cerf proved the pseudoisotopy theorem which implies that is the trivial group provided , and so provided . Applications If M is a piecewise linear manifold then the problem of finding the compatible smooth structures on M depends on knowledge of the groups . More precisely, the obstructions to the existence of any smooth structure lie in the groups for various values of k, while if such a smooth structure exists then all such smooth structures can be classified using the groups . In particular the groups Γk vanish if , so all PL manifolds of dimension at most 7 have a smooth structure, which is essentially unique if the manifold has dimension at most 6. The following finite abelian groups are essentially the same: The group Θn of h-cobordism classes of oriented homotopy n-spheres. The group of h-cobordism classes of oriented n-spheres. The group Γn of twisted oriented n-spheres. The homotopy group n(PL/DIFF) If , the homotopy group n(TOP/DIFF) (if this group has order 2; see Kirby–Siebenmann invariant). The group of smooth structures of an oriented PL n-sphere. If , the group of smooth structures of