  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 $S^2 \times S^2$, is
smoothly isotopic to a diffeomorphism that is supported on a contractible
submanifold. For those that require more than one copy of $S^2 \times S^2$, 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.
