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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Resolutions for metrizable compacta in extension theory

Authors: Leonard R. Rubin and Philip J. Schapiro
Journal: Trans. Amer. Math. Soc. 358 (2006), 2507-2536
MSC (2000): Primary 55P55, 54F45
Published electronically: May 26, 2005
MathSciNet review: 2204042
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Abstract: We prove a $K$-resolution theorem for simply connected CW- complexes $K$ in extension theory in the class of metrizable compacta $X$. This means that if $K$ is a connected CW-complex, $G$ is an abelian group, $n\in \mathbb N _{\geq 2}$, $G=\pi _{n}(K)$, $\pi _{k}(K)=0$ for $0\leq k<n$, and $\operatorname{extdim} X\leq K$ (in the sense of extension theory, that is, $K$ is an absolute extensor for $X$), then there exists a metrizable compactum $Z$ and a surjective map $\pi :Z\rightarrow X$ such that:

(a) $\pi $ is $G$-acyclic,

(b) $\dim Z\leq n+1$, and

(c) $\operatorname{extdim} Z\leq K$.

This implies the $G$-resolution theorem for arbitrary abelian groups $G$ for cohomological dimension $\dim _{G} X\leq n$ when $n\in \mathbb N_{\geq 2}$. Thus, in case $K$ is an Eilenberg-MacLane complex of type $K(G,n)$, then (c) becomes $\dim _{G} Z\leq n$.

If in addition $\pi _{n+1}(K)=0$, then (a) can be replaced by the stronger statement,

(aa) $\pi $ is $K$-acyclic.

To say that a map $\pi $ is $K$-acyclic means that for each $x\in X$, every map of the fiber $\pi ^{-1}(x)$ to $K$ is nullhomotopic.

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Additional Information

Leonard R. Rubin
Affiliation: Department of Mathematics, University of Oklahoma, 601 Elm Ave., Rm. 423, Norman, Oklahoma 73019

Philip J. Schapiro
Affiliation: Department of Mathematics, Langston University, Langston, Oklahoma 73050

Keywords: Bockstein basis, Bockstein inequalities, \v{C}ech cohomology, cell-like map, cohomological dimension, CW-complex, dimension, Edwards-Walsh resolution, Eilenberg-Mac\, Lane complex, $G$-acyclic resolution, inverse sequence, $K$-acyclic resolution, Moore space, shape of a point, simplicial complex
Received by editor(s): March 13, 2002
Received by editor(s) in revised form: May 11, 2004
Published electronically: May 26, 2005
Article copyright: © Copyright 2005 American Mathematical Society