Non cell like decompositions of \(S^ 3\)
The authors consider two classes of maps, \(f: S^ 3\to X\), from the 3- sphere to a compactum X. The first class consists of maps that are non- constant approximate fibrations whose fiber is a pointed FANR. They establish that either f factors as \(f=pg\) where \(p: Y\to X\) is a covering map and g is cell like or f can be approximated by maps topologically equivalent to the Hopf fibration. The second class consists of maps with the property that \(f^{-1}(x)\) has the shape of a circle for each \(x\in X\). For such maps they establish that if X is an ANR and each point inverse links some other point inverse, then \(X\approx S^ 2\). [Recently, the reviewer and \textit{R. J. Daverman} have extended this analysis and removed both the assumption that X is an ANR and that point inverses link; see Trans. Am. Math. Soc. 288, 273-291 (1985).]
maps with fibre of the shape of a circle
scientific article; zbMATH DE number 3879730