Milnor's exotic sphere

Abstract

This thesis follows the construction by Milnor of a 7-manifold that is homeomorphic to the 7-sphere but not diffeomorphic. By gluing two solid tori along the boundary under a diffeomorphism you obtain a 7-manifold. Points on these tori can be described in quaternions. By using the behaviour of quaternions we can find the desired diffeomorphisms, which we use for identifying points of the two tori. To show that this construction is homeomorphic we use tools from Morse theory. In particular, we use Reeb's lemma which says that if there exist a Morse funtion with only two critical points on a n-manifold, it is homeomorphic to the n-sphere but may not diffeomorphic. Milnor used results from Thoms to create an invariant which measures somehow how far off the manifold is of being the boundary of an 8-manifold. Since the 7-sphere is the boundary of the 8-disk, this would yield 0. But the construction of the 7-manifold homeomorphic to the 7-sphere yields another results, which indicates, it is not diffeomorphic.