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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.

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Since dim(H_2(M)) gives the number of non-homotopic 2-spheres which cannot be contracted to a point, I suspect that, for the case S=S^2, the number of topological inequivalent manifold that one can get is at least dim(H_2(M))+1, where the +1 correspond to removing a "trivial" sphere. Maybe there are more though, depending on how rich is the topology of the "hole" that prevents one of the generators of H_2(M) to be contracted to a point. – GFR Mar 3 '13 at 19:12

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