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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cohomological dimension
and metrizable spaces. II


Author: Jerzy Dydak
Journal: Trans. Amer. Math. Soc. 348 (1996), 1647-1661
MSC (1991): Primary 55M11, 54F45
MathSciNet review: 1333390
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Abstract: The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$.

Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$.

As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension:

Theorem. Suppose $A,B$ are subsets of a metrizable space. Then

\begin{equation*}\dim _{{\mathbf R} }(A\cup B)\le \dim _{{\mathbf R} }A+\dim _{{\mathbf R} }B+1 \end{equation*}

for any ring ${\mathbf R} $ with unity and

\begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*}

for any abelian group $G$.

The second part of the paper is devoted to the question of existence of universal spaces:

Theorem. Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then

a.
Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$.
b.
There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$.
c.
There is a completely metrizable and separable space $Z$ such that $ K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z'$ with $ K_{i}\in AE(Z')$ for all $i\ge 1$ embeds in $Z$ as a closed subset.


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Additional Information

Jerzy Dydak
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: dydak@math.utk.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01536-X
PII: S 0002-9947(96)01536-X
Keywords: Dimension, cohomological dimension, Menger-Urysohn Theorem, absolute extensors, Eilenberg--Mac Lane spaces, universal spaces, compactifications
Received by editor(s): December 11, 1992
Received by editor(s) in revised form: May 3, 1995
Additional Notes: Supported in part by a grant from the National Science Foundation
Article copyright: © Copyright 1996 American Mathematical Society