\(G\)-\(s\)-cobordant manifolds are not necessarily \(G\)-homeomorphic for arbitrary compact Lie groups \(G\)
type: Item
instance of: scholarly article
title: \(G\)-\(s\)-cobordant manifolds are not necessarily \(G\)-homeomorphic for arbitrary compact Lie groups \(G\)
zbMATH Open document ID: 0818.57021
author: Katsuo Kawakubo
published in: Journal of the Mathematical Society of Japan
publication date: 1994-06-16T00:00:00Z
review text: \textit{S. Araki} and the author [ibid. 40, No. 2, 349-367 (1988; Zbl 0654.57019)] presented an equivariant version of the classical \(s\)- cobordism theorem as follows. Let \(G\) be a compact Lie group and let \((W;X,Y)\) be a \(G\)-equivariant \(h\)-cobordism triad of smooth \(G\)- manifolds fulfilling a suitable gap hypothesis. If the \(G\)-equivariant Whitehead torsion \(\tau_ G (i_ X)\) of the inclusion \(i_ X : X \to W\) vanishes, then \(W\) and \(X \times [0,1]\) are \(G\)-diffeomorphic rel \(X\), and hence the manifolds \(X\) and \(Y\) are \(G\)-diffeomorphic. In his article [Lect. Notes Math. 1375, 183-190 (1989; Zbl 0671.57022)], the author of the paper under review shows that the result does not hold without assuming the gap hypothesis by constructing examples of \(G\)-equivariant \(h\)-cobordism triads \((W;X,Y)\) with \(\tau_ G (i_ X) = 0\), in which \(X\) and \(Y\) are \(G\)-diffeomorphic but \(W\) and \(X \times [0,1]\) are not. In the paper under review, the author shows in turn that without assuming the gap hypothesis, it may also happen that \(\tau_ G (i_ X) = 0\) but the manifolds \(X\) and \(Y\) are not \(G\)-homeomorphic, and in particular, \(W\) and \(X \times [0,1]\) are not \(G\)-homeomorphic. The paper contains complete information about related results obtained by other authors.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57R80
Mathematics Subject Classification ID: 57R91
zbMATH DE Number: 562713
zbMATH Keywords: \(G\)-equivariant \(s\)-cobordism theorem
zbMATH Keywords: \(G\)-equivariant Whitehead torsion
zbMATH Keywords: \(G\)-equivariant \(h\)-cobordism triads
zbMATH Keywords: gap hypothesis
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04540599
OpenAlex ID: W2094793094
Wikidata QID: Q115225510
DOI: 10.2969/JMSJ/04540599
description: scientific article; zbMATH DE number 562713