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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
DOI: https://doi.org/10.1090/S0002-9947-96-01536-X
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:

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

    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