Pseudo-isotopies of 3-manifolds
A homotopy \(H:X \times [0,1]\to Y\) between homeomorphisms \(f_0\), \(f_1:X \to Y\) is an isotopy, if it is a homeomorphism for every \(t\in[0,1]\). A homeomorphism \(H_\sharp: X\times[0,1] \to Y\times [0,1]\) is a pseudo-isotopy between homeomorphisms \(f_0\), \(f_1: X\to Y\) if \(H_\sharp (x,0)=(f_0(x),0)\) and \(H_\sharp (x,1) = (f_1(x),1)\). If \(X\) is a locally compact Hausdorff space, then isotopic homeomorphisms are pseudo-isotopic. Therefore, for manifolds isotopy implies pseudoisotopy. It also holds in the opposite direction for 1-dimensional, 2-dimensional, some classes of irreducible 3-manifolds and simply connected manifolds of dimension greater than three. The converse implication does not hold in general since in higher dimensions there are pseudo-isotopic but not isotopic homeomorphisms, like \(n\)-dimensional torus \(T^n\), \(n\geq 4\). The present paper treats this opposite implication for 3-dimensional manifolds and finds the negative answer among reducible 3-manifolds, in particular among metacyclic prism manifolds (3-dimensional spherical space whose fundamental group is properly metacyclic i.e. nonabelian with a cyclic Sylow 2-subgroup). We only quote the result for closed 3-manifolds: If \(M^3\) and \(N^3\) are metacyclic prism manifolds, then the connected sum \(M^3 \sharp N^3\) admits homeomorphisms that are pseudo-isotopic but not isotopic to the identity. The homeomorphisms in question are examples of homeomorphisms considered in [\textit{J. L. Friedman} and \textit{D. M. Witt}, ibid. 25, 35-44 (1986; Zbl 0596.57008)].
scientific article; zbMATH DE number 878547