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On a problem of P. S. Aleksandrov. (Russian) Zbl 0643.55001
This outstanding paper gives an affirmative answer to one of the major unsolved problems of geometric topology, whether or not there is a dimension raising cell-like map. Namely, the author constructs an example of an infinite-dimensional compact metric space X with finite (integral) cohomological dimension: c-dim X\(=3\). Thus he solves a classical problem from dimension theory, formulated over half a century ago by P. S. Aleksandrov [Mat. Sb., Nov. Ser. 1(43), 619-634 (1937; Zbl 0015.37502)], which for decades seemed to be hopelessly difficult. Equally important, however, is that the existence of such an example implies, by the well- known work of R. D. Edwards [exposed in the survey by J. J. Walsh, Dimension, cohomological dimension and cell-like mappings. Shape theory and geometric topology, Proc. Conf. Dubrovnik 1981, Lect. Notes Math. 870, 105-118 (1981; Zbl 0474.55002)] that for every integer \(n\geq 7\) there is a proper cell-like map of the Euclidean n-space onto an infinite-dimensional space. Therefore cell-like maps can actually raise dimension!
(Reviewer’s remark: It remains unknown if cell-like maps can also raise dimension on topological n-manifolds of dimension \(n=4\), 5, or 6, whereas it was shown earlier that such a phenomenon is not possible for \(n\leq 3\)- see the survey by W. J. R. Mitchell and the reviewer [Topology of cell-like mappings, Rend. Sem. Fac. Sci. Univ. Cagliari, to appear].)
Reviewer: D.Repovš

MSC:
55M10 Dimension theory in algebraic topology
54F45 Dimension theory in general topology
57N25 Shapes (aspects of topological manifolds)
57N60 Cellularity in topological manifolds
54C56 Shape theory in general topology
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