Diffeotopy Groups of Noncompact 4-Manifolds

We provide information on diffeotopy groups of exotic smoothings of punctured 4-manifolds, extending previous results on diffeotopy groups of exotic R4s. In particular, we prove that for a smoothable 4-manifold M and for a nonempty discrete set of points S⊊M˚, there are uncountably many distinct smoothings of M∖S whose diffeotopy groups are uncountable. We then prove that for a smoothable 4-manifold M and for a nonempty discrete set of points S⊊M˚, there exists a smoothing of M∖S whose diffeotopy groups have similar properties as RU, Freedman and Taylor’s universal R4. Moreover, we prove that if M is nonsmoothable, then both results still hold under the assumption that |S|≥2.