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Dimension of compact metric spaces. (English) Zbl 1382.55002

In the paper under review, the author gives a survey of classical and new results in dimension theory of compact metric spaces. In the first part, the author recalls classical and recently obtained results from covering dimension theory. Topics include the dimension of unions, the dimension of products, the Hurewicz mapping theorem, dimension raising maps, stable intersection in \({\mathbb R}^n\), and universal spaces.
In the second part, he recalls basic results in cohomological dimension theory: the Alexandroff Theorem, characterization of cohomological dimension in terms of cell-like or acyclic maps, Bockstein theory, and Shchepin’s arithmetics. He also lists important results on infinite dimensional compacta, in particular, an approach based on generalized cohomology. In the last part, he sketches some applications of dimension theory to topology of manifolds and geometric group theory: convergence in Gromov-Hausdorff spaces, surgery on topological manifolds, Hilbert-Smith conjecture, boundaries of groups, and large scale dimension.

MSC:

55M10 Dimension theory in algebraic topology
54F45 Dimension theory in general topology
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
54-02 Research exposition (monographs, survey articles) pertaining to general topology
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