Extension theory of infinite symmetric products

Tom 182 / 2004

Jerzy Dydak Fundamenta Mathematicae 182 (2004), 53-77 MSC: 54F45, 55M10, 55N99, 55Q40, 55P20. DOI: 10.4064/fm182-1-3


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 $\mathop {\rm ext\hbox {-}dim}\nolimits (X)$ 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 $\mathop {\rm SP}\nolimits (L)$. One of the main ideas of the paper is to treat $\mathop {\rm ext\hbox {-}dim}\nolimits (X)\leq \mathop {\rm SP}\nolimits (L)$ as the fundamental concept of cohomological dimension theory instead of $\mathop {\rm dim}\nolimits _G(X)\leq n$. 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 $\mathop {\rm SP}\nolimits (L)^X$). 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.


  • Jerzy DydakMathematics Department
    University of Tennessee
    Knoxville, TN 37996-1300, U.S.A.

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