Delooping controlled pseudo-isotopies of Hilbert cube manifolds
Let \(p: E\to B\) be a Hurewicz fibration, E a compact Hilbert cube manifold, and B a compact polyhedron (or a compact TOP manifold with a handle decomposition). The author defines the controlled Whitehead [respectively, pseudo-isotopy] ''space'' Wh(p) [respectively, \({\mathcal P}(p)]\). These ''spaces'' are semisimplicial complexes. It is proved that \(\Omega\) (Wh(p))\(\simeq {\mathcal P}(p)\), and that \(\pi_ 0(Wh(p))\) is the appropriate obstruction group for deciding whether a ''controlled'' homotopy equivalence \(f: M\to E\) between compact Hilbert cube manifolds is \(p^{-1}(\epsilon)\) homotopic to a homeomorphism for every \(\epsilon >0\). Here ''control'' means arbitrarily small \(\epsilon\) control in B.
scientific article; zbMATH DE number 4009339
scientific article