Extendable diffeomorphisms in the 4-sphere

A diffeomorphism over an embedded surface is called extendable if it can extend to a diffeomorphism of the 4-manifold that the surface is embedded in. Extendable diffeomorphisms form a subgroup of the mapping class group of a surface, a number of which were computed by Susumu Hirose in the early 2000's. We'll review Hirose's techniques and see why certain 4-manifolds, such as CP^2, are easier to extend in than others, such as the 4-sphere. Then we'll review more recent extendability results in S^4, all of which rely on Hirose's seminal work.