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

Miyata, Takahisa

doi:10.1090/S0002-9939-01-05852-X

Shape aspherical compacta–applications of a theorem of Kan and Thurston to cohomological dimension and shape theories
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Proc. Amer. Math. Soc. 129 (2001), 2783-2788 Request permission

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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