On low dimensional s-cobordisms
type: Item
instance of: scholarly article
title: On low dimensional s-cobordisms
zbMATH Open document ID: 0605.57017
DOI: 10.1007/BF02621925
author: Slawomir Kwasik
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140061
review text: This paper proves a 4-dimensional s-cobordism theorem for 3-manifolds with nontrivial polycyclic fundamental group. The significant new tool in this proof is \textit{M. H. Freedman}'s proof of the 5-dimensional s- cobordism theorem for polyfinite and polycyclic groups [Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 647-663 (1984; Zbl 0577.57003)]. The author also shows, by constructing a nontrivial 4-dimensional s-cobordism between two copies of \(S^ 1\times {\mathbb{R}}P^ 2\), that the 4- dimensional s-cobordism theorem cannot hold for all groups covered by Freedman's theorem. As noted, this can also be deduced from \textit{T. Matumoto} and \textit{L. Siebenmann} [Math. Proc. Camb. Philos. Soc. 84, 85-87 (1978; Zbl 0411.57034)]. Orientable examples have been given by \textit{S. E. Cappell} and \textit{J. L. Shaneson} [J. Differ. Geom. 22, 97-115 (1985; Zbl 0597.57008)]. The author also points out that his theorem and constructions show that a manifold homotopy equivalent to \(S^ 3\times S^ 1\), \(S^ 2\times T^ 2\), and \(T^ 4\) is homeomorphic to the corresponding manifold. Furthermore he utilizes his constructions to show the existence of a fake topological \({\mathbb{R}}P^ 4\), first constructed by \textit{D. Ruberman} [Topology Appl. 18, 217-224 (1984; Zbl 0559.57016)].
Mathematics Subject Classification ID: 57R80
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 3977951
zbMATH Keywords: 4-dimensional s-cobordism theorem for 3-manifolds
zbMATH Keywords: polycyclic fundamental group
zbMATH Keywords: homotopy equivalent
zbMATH Keywords: homeomorphic
zbMATH Keywords: fake topological \({bbfR}P^ 4\)
reviewed by: R. J. Stern
MaRDI profile type: Publication
OpenAlex ID: W2321306826
description: scientific article; zbMATH DE number 3977951