Extension theory of infinite symmetric products
| dc.creator | Dydak, Jerzy | |
| dc.date | 2004-04-19 | |
| dc.date.accessioned | 2026-07-07T09:23:24Z | |
| dc.date.available | 2026-07-07T09:23:24Z | |
| dc.description | 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 $\ExD(X)$ was introduced by A.N.Dranishnikov \cite {D$_5$} 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 $\ExD(X)\leq SP(L)$ as the fundamental concept of cohomological dimension theory instead of $\dim_G(X)\leq n$. In a subsequent paper \cite{Dy$_6$} we show how properties of infinite symmetric products lead naturally to a calculus of graded groups which implies most of classical results of the cohomological dimension. The basic notion in \cite{Dy$_6$} is that of homological dimension of a graded group which allows for simultanous 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 characterization of infinite symmetric products of the same extension type as a compact (or finite-dimensional and countable) CW complex. | |
| dc.description | To appear in Fundamenta Mathematicae | |
| dc.identifier | https://arxiv.org/abs/math/0404332 | |
| dc.identifier | http://arxiv.org/abs/math/0404332 | |
| dc.identifier | Fund.Math. 182 (2004), 53-77 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155719 | |
| dc.subject | Algebraic Topology | |
| dc.subject | General Topology | |
| dc.subject | Geometric Topology | |
| dc.subject | 54F45, 55M10, 55N99, 55Q40, 55P20 | |
| dc.title | Extension theory of infinite symmetric products | |
| dc.type | text |