Extension theory of infinite symmetric products

Jerzy Dydak

Fundamenta Mathematicae (2004)

  • Volume: 182, Issue: 1, page 53-77
  • ISSN: 0016-2736

Abstract

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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 d i m G ( X ) ≤ n . I n a s u b s e q u e n t p a p e r [ 18 ] w e s h o w h o w p r o p e r t i e s o f i n f i n i t e s y m m e t r i c p r o d u c t s l e a d n a t u r a l l y t o a c a l c u l u s o f g r a d e d g r o u p s w h i c h i m p l i e s m o s t o f t h e c l a s s i c a l r e s u l t s o n t h e c o h o m o l o g i c a l d i m e n s i o n . T h e b a s i c n o t i o n i n [ 18 ] i s t h a t o f h o m o l o g i c a l d i m e n s i o n o f a g r a d e d g r o u p w h i c h a l l o w s f o r s i m u l t a n e o u s t r e a t m e n t o f c o h o m o l o g i c a l d i m e n s i o n o f c o m p a c t a a n d e x t e n s i o n p r o p e r t i e s o f C W c o m p l e x e s . We introduce cohomology of X with respect to L (defined as homotopy groups of the function space S P ( 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.

How to cite

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Jerzy Dydak. "Extension theory of infinite symmetric products." Fundamenta Mathematicae 182.1 (2004): 53-77. <http://eudml.org/doc/283003>.

@article{JerzyDydak2004,
abstract = {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.},
author = {Jerzy Dydak},
journal = {Fundamenta Mathematicae},
keywords = {cohomological dimension; dimension; Eilenberg-MacLane complexes; dimension functions; infinite symmetric product},
language = {eng},
number = {1},
pages = {53-77},
title = {Extension theory of infinite symmetric products},
url = {http://eudml.org/doc/283003},
volume = {182},
year = {2004},
}

TY - JOUR
AU - Jerzy Dydak
TI - Extension theory of infinite symmetric products
JO - Fundamenta Mathematicae
PY - 2004
VL - 182
IS - 1
SP - 53
EP - 77
AB - 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.
LA - eng
KW - cohomological dimension; dimension; Eilenberg-MacLane complexes; dimension functions; infinite symmetric product
UR - http://eudml.org/doc/283003
ER -

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