The study of exotic smooth structures on manifolds is one of the fundamental problems in topology. In particular, the classification of smooth structures on a given smooth manifold M is connected to the determination of a subgroup of the group of homotopy spheres, namely, the concordance inertia group of M. In this talk, we compute the concordance inertia group of the product of a closed, smooth 4-manifold M with the standard k-sphere using the stable homotopy type of M, where k varies between 1 to 10.
Using the above computations of the concordance inertia group, we classify all smooth manifolds homeomorphic to the product of M with the standard k-spheres, up to concordance. As an application of the above computations, we give a complete diffeomorphic classification of all closed, oriented, smooth manifolds homeomorphic to the product of a complex 2 projective space with standard k-sphere, where k lies between 4 and 6.
