Examples of cell-like maps that are not shape equivalences (Q790484)

\textit{J. L. Taylor} constructed in the early 1970's a cell-like map F from a compactum T with nontrivial shape onto the Hilbert cube Q [Bull. Am. Math. Soc. 81, 629-632 (1975; Zbl 0316.55010)]. The authors make a careful geometric analysis of the Taylor example in order to answer other standard but previously unresolved questions in decomposition theory. As they state in their introduction, they construct (1) a cell-like map H from the compactum T onto Q such that the nondegeneracy set \(N_ H=\{q\in Q:\quad H^{-1}(q)\neq point\}\) is a countable union of finite dimensional compacta; (2) a map from a compactum onto Q whose point inverses are finite dimensional absolute retracts and which is not a shape equivalence; and (3) a locally contractible compactum Z which is not an ANR, compacta X and Y which are ANR's and cell-like maps g: \(X\to Z\) and f: \(Z\to Y\) with the property that \(N_ g\cap f^{-1}(N_ f)=\emptyset\).