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2004 | 182 | 1 | 53-77
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Extension theory of infinite symmetric products

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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 ext-dim(X) was introduced by A. N. Dranishnikov [9] in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products SP(L). One of the main ideas of the paper is to treat ext-dim(X) ≤ SP(L) as the fundamental concept of cohomological dimension theory instead of $dim_G(X) ≤ n. In a subsequent paper [18] 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 [18] 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 $SP(L)^X$). 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.
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  • Mathematics Department, University of Tennessee, Knoxville, TN 37996-1300, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-1-3
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