On smoothable surgery for 4–manifolds

Abstract

Under certain homological hypotheses on a compact 4–manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4–manifolds with certain product geometries. Most of these compact manifolds have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman–Quinn topological surgery. Necessarily, the $\ast$–construction of certain non-smoothable homotopy equivalences requires surgery on topologically embedded 2–spheres and is not attacked here by transversality and cobordism.