Surfaces in 4-manifolds and their mapping class groups

A surface in a smooth 4-manifold is called flexible   if, for any diffeomorphism ϕϕ on the surface, there is a diffeomorphism on the 4-manifold whose restriction on the surface is ϕϕ and which is isotopic to the identity. We investigate a sufficient condition for a smooth 4-manifold MM to include flexible knotted surfaces, and introduce a local operation in simply connected 4-manifolds for obtaining a flexible knotted surface from any knotted surface.