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

DOI:
https://doi.org/10.1090/S0002-9947-05-03747-5

Published electronically:
May 26, 2005

MathSciNet review:
2204042

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to 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

Email:
lrubin@ou.edu

**Philip J. Schapiro**

Affiliation:
Department of Mathematics, Langston University, Langston, Oklahoma 73050

Email:
pjschapiro@lunet.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03747-5

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