Corks for exotic diffeomorphisms

We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply connected 4-manifold that is isotopic to the identity after stabilizing with one copy of S 2 S 2 is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold.For those that require more than one copy of S 2 S 2 , we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map.We investigate the implications of these results by applying them to known exotic diffeomorphisms.