On Surgery for 4-manifolds and Splitting of 5-manifolds

Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. This technical result allows an extension of Cappell’s 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Typically, these homotopy fibers 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.