\(G\)-\(s\)-cobordant manifolds are not necessarily \(G\)-homeomorphic for arbitrary compact Lie groups \(G\)
\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.
\(G\)-equivariant \(s\)-cobordism theorem
\(G\)-equivariant \(h\)-cobordism triads
scientific article; zbMATH DE number 562713