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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Shape aspherical compacta-applications of a theorem of Kan and Thurston to cohomological dimension and shape theories


Author: Takahisa Miyata
Journal: Proc. Amer. Math. Soc. 129 (2001), 2783-2788
MSC (1991): Primary 55M10, 55P55; Secondary 54F45, 55N05
Published electronically: January 18, 2001
MathSciNet review: 1838803
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Abstract:

Dydak and Yokoi introduced the notion of shape aspherical compactum. In this paper, we use this notion to obtain a generalization of Kan and Thurston theorem for compacta and pro-homology. As an application, we obtain a characterization of cohomological dimension with coefficients in $\mathbb{Z}$ and $\mathbb{Z}/p$ ($p$ prime) in terms of acyclic maps from a shape aspherical compactum, which improves the theorems of Edwards and Dranishnikov. Furthermore, we obtain the shape version of the theorem and as a consequence we show that every compactum has the stable shape type of a shape aspherical compactum.


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Additional Information

Takahisa Miyata
Affiliation: Department of Computer Science, Shizuoka Institute of Science and Technology, 2200-2 Toyosawa, Fukuroi, Shizuoka-Pref., 437-8555 Japan
Email: miyata@mb.sist.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05852-X
PII: S 0002-9939(01)05852-X
Keywords: Shape aspherical compactum, approximately aspherical compactum, cohomological dimension, shape, Kan and Thurston theorem
Received by editor(s): August 16, 1999
Received by editor(s) in revised form: December 29, 1999
Published electronically: January 18, 2001
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2001 American Mathematical Society