an oriented topological n-sphere. If , the group of components of the group of all orientation-preserving diffeomorphisms of Sn−1. 4-dimensional exotic spheres and Gluck twists In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincaré conjecture", and is discussed by who say that it is believed to be false. Some candidates proposed for exotic 4-spheres are the Cappell–Shaneson spheres () and those derived by Gluck twists . Gluck twist spheres are constructed by cutting out a tubular neighborhood of a 2-sphere S in S4 and gluing it back in using a diffeomorphism of its boundary S2×S1. The result is always homeomorphic to S4. Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture. For example, , , , , , , , . See also Milnor's sphere Atlas (topology) Clutching construction Exotic R4 Cerf theory Seven-dimensional space References This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds. – This paper describes the structure of the group of smooth structures on an n-sphere for n > 4. The promised paper "Groups of Homotopy Spheres: II" never appeared, but Levine's lecture notes contain the material which it might have been expected to contain. . . . External links Exotic spheres on the Manifold Atlas Exotic sphere home page on the home page of Andrew Ranicki. Assorted source material relating to exotic spheres. An animation of exotic 7-spheres Video from a presentation by Niles Johnson at the Second Abel conference in honor of John Milnor. The Gluck construction on the Manifold Atlas Differential topology Differential structures Surgery theory Spheres