Dual decompositions of 4-manifolds III: s-cobordisms

Abstract

The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a “core” sub-s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-complex. The decomposition is highly nonunique so cannot be used to define an invariant, but it shows that the topological s-cobordism question reduces to the core case. The simply-connected version of the decomposition (with 1-complex a point) is due to Curtis, Freedman, Hsiang and Stong. Controlled surgery is used to reduce topological triviality of core s-cobordisms to a question about controlled homotopy equivalence of 4-manifolds. There are speculations about further reductions. The decompositions on the ends of the s-cobordism are “dual decompositions” with homotopically-controlled handle structures, and the main result is an application of earlier papers in the series.