Extension theory of infinite symmetric products

Jerzy Dydak¹

¹ Mathematics Department University of Tennessee Knoxville, TN 37996-1300, U.S.A.

Fundamenta Mathematicae, Tome 182 (2004) no. 1, pp. 53-77.

Résumé

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.

Détail

DOI : 10.4064/fm182-1-3

Keywords: present approach cohomological dimension theory based infinite symmetric products general theory dimension called extension dimension notion extension dimension mathop ext hbox dim nolimits introduced dranishnikov context compact spaces complexes paper investigates extension types infinite symmetric products mathop nolimits main ideas paper treat mathop ext hbox dim nolimits leq mathop nolimits fundamental concept cohomological dimension theory instead mathop dim nolimits leq subsequent paper properties infinite symmetric products lead naturally calculus graded groups which implies classical results cohomological dimension basic notion homological dimension graded group which allows simultaneous treatment cohomological dimension compacta extension properties complexes introduce cohomology respect defined homotopy groups function space mathop nolimits application results characterize countable groups moore space extension type eilenberg maclane space another application characterization infinite symmetric products extension type compact finite dimensional countable complex