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Extension theory of infinite symmetric products. (English) Zbl 1072.55001
Summary: We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension $$\text{ext-dim}(X)$$ was introduced by A. N. Dranishnikov [Math. USSR, Sb. 74, 47-56 (1993; Zbl 0774.55011)] in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products $$\text{SP}(L)$$. One of the main ideas of the paper is to treat $$\text{ext-dim}(X) \leq \text{SP}(L)$$ as the fundamental concept of cohomological dimension theory instead of $$\dim_G(X)\leq n$$. In a subsequent paper [Algebra of dimension theory, Trans. Am. Math. Soc., to appear] we show how properties of infinite symmetric products lead naturally to a calculus of graded groups which implies most of the classical results on the cohomological dimension. The basic notion in [Loc. cit.] is that of homological dimension of a graded group which allows for simultaneous treatment of cohomological dimension of compacta and extension properties of CW complexes. We introduce cohomology of $$X$$ with respect to $$L$$ (defined as homotopy groups of the function space $$\text{SP}(L)^X$$). As an application of our results we characterize all countable groups $$G$$ so that the Moore space $$M(G,n)$$ is of the same extension type as the Eilenberg-MacLane space $$K(G,n)$$. Another application is a characterization of infinite symmetric products of the same extension type as a compact (or finite-dimensional and countable) CW complex.

##### MSC:
 55M10 Dimension theory in algebraic topology 54F45 Dimension theory in general topology 55P20 Eilenberg-Mac Lane spaces 54C20 Extension of maps
Zbl 0774.55011
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