The stable topology of 4-manifolds
type: Item
instance of: scholarly article
title: The stable topology of 4-manifolds
zbMATH Open document ID: 0536.57006
DOI: 10.1016/0166-8641(83)90049-4
author: Frank Quinn
published in: Topology and its Applications
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R80
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57R95
zbMATH DE Number: 3852031
zbMATH Keywords: Casson move
zbMATH Keywords: stable theory of smooth 4-manifolds
zbMATH Keywords: transfer families of 2- spheres
zbMATH Keywords: Norman trick
zbMATH Keywords: stable 5-dimensional s-cobordism theorem
zbMATH Keywords: 1- connected smooth 4-manifold pairs
reviewed by: R. J. Stern
MaRDI profile type: Publication
full work available at URL: 0166-8641(83)90049-4
OpenAlex ID: W1979468547
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description: scientific article; zbMATH DE number 3852031