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Fiber shape theory, shape fibrations and movability of maps. (English) Zbl 0632.55009

Geometric topology and shape theory, Proc. Conf., Dubrovnik/Yugosl. 1986, Lect. Notes Math. 1283, 240-252 (1987).
[For the entire collection see Zbl 0619.00017.]
This is a very informative and rather complete survey on fiber shape theory, shape fibrations and related notions. The author considers locally compact separable metric spaces and proper maps and uses the definition of a shape fibration as given by the reviewer [Fund. Math. 114, 53-78 (1981; Zbl 0411.54019)]. The survey covers approximate homotopy lifting properties, the fiber shape category, global and local movability of mappings and the fiber shape of Q-manifold bundles. The author also announces some new results. In particular, if \(f: X\to Y\) is a weak shape fibration, X is connected, Y is \(UV^ 1\) and the fibers F of f are connected FANR’s, then there exists a spectral sequence with \(E_ 2^{pq}=pro-H_ p(Y;H_ q(F))\Rightarrow pro-H_{p+q}(X)\). The Leray spectral sequence (for Čech cohomology) is fiber shape invariant. The bibliography consists of 57 items.
Reviewer: S.Mardešić

MSC:

55R65 Generalizations of fiber spaces and bundles in algebraic topology
55T10 Serre spectral sequences
57N20 Topology of infinite-dimensional manifolds
55P55 Shape theory