An exotic sphere is a differentiable manifold which is homeomorphic to a sphere but not diffeomorphic. That in fact exotic spheres exist is due to John Milnor who showed in 1956 that there are at ... read more
An exotic sphere is a differentiable manifold which is homeomorphic to a sphere but not diffeomorphic. That in fact exotic spheres exist is due to John Milnor who showed in 1956 that there are at least 10 exotic 7-spheres, i. e. that there are at least 11 pairwise non-diffeomorphic smooth manifolds, all homeomorphic to the standard 7-sphere. At that time this was a big surprise.
In the seminar we have a look at a real classic, the paper of Kervaire and Milnor from 1963 where most of what is known today about the structure of the set of homotopy n-spheres, i.e. the set of smooth manifolds homotopy equivalent to the standard n-sphere, was established.
Here we can see how all these things that we learned in the algebraic courses and some more homotopy theory can be applied to obtain deep and surprisingly far reaching results about the classification of smooth manifolds homeomorphic to spheres, and that it is the lack of knowledge about the stable homotopy groups of spheres that prevent us from giving the complete answer.
The seminar should be self contained with prerequisites being basic differential and algebraic topology. Depending on the knowledge of the participants we might include some preparatory talks summarazing these basics. The seminar however is planned as a companion to the lecture course on surgery theory and it is highly recommended to attend both.
Kervaire, Milnor: Groups of homotopy spheres I, Annals of Math. 77, 504-537 (1963).