Topological inequivalent manifolds obtaining by removing a surface from a manifold

Question

Are there any general techniques for classifying the inequivalent topologies that can be obtained by removing a 2-surface S from a 4-manifold M? I am particularly interest in the case where both M and S are compact and smooth. A simple example would be the removal of a Riemann surface from CP^2. I would then be interested in how one could do the same for the connected sum of various CP^2.