The stable topology of 4-manifolds
This paper presents several results concerning the stable (i.e. connected sums with \(S^ 2\times S^ 2)\) theory of smooth 4-manifolds that revolve around the study of transfer families of 2-spheres. Given a collection of 2-spheres in a smooth 4-manifold \(M^ 4\), the Norman trick allows the geometric cancellation of algebraically cancelling pairs of intersections provided there is a framed embedded transverse family of 2- spheres. The author then presents an algebraic characterization of collections of 2-spheres in \(M^ 4\) which have a framed embedded transverse family in some stabilization M{\#}k\((S{}^ 2\times S^ 2)\). Applying this and the Casson technology to improve the homotopy behavior of the complements of embedded spheres, the author deduces new stable theorems; in particular a stable 5-dimensional s-cobordism theorem and the fact that 1-connected smooth 4-manifold pairs stably have handle decompositions without 1-handles. This paper was written before \textit{M. H. Freedman}'s proof of the 4-dimensional topological Poincaré conjecture. The author points out that the embedding theorems for towers of immersed disks presented by Freedman [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)], can be proven by recentlessly utilizing the techniques of this paper under review.
stable 5-dimensional s-cobordism theorem
scientific article; zbMATH DE number 3852031
scientific article