Shape aspherical compacta -- applications of a theorem of Kan and Thurston to cohomological dimension and shape theories (Q2716123)

The Kan-Thurston theorem says that given a path-connected space \(X\), there exists a \(K(\pi, 1)\)-space \(Y\) and a map \(t:Y \to X\), natural for maps on \(X\) such that \(t\) induces isomorphisms in singular homology as well as in cohomology with local coefficients, and in the fundamental groups \( t_*:\pi_1(Y) \to \pi_1(X)\) is onto. Using the proof and a variation of the above theorem given by \textit{C. F. Maunder}, the author obtains the following generalization of the Kan-Thurston theorem: NEWLINENEWLINENEWLINEFor each continuum \(X\), there exists an approximately aspherical compactum \(Y\) with \(\dim Y= \dim X\), and a surjective map \(\varphi :Y\to X\) such that \(\varphi\) induces isomorphisms in Čech homologies and cohomologies and \(\varphi_* : \text{pro-}\pi_1^S(Y)\to \text{pro-}\pi_1^S(X)\) is an epimorphism; in addition, the map \(\varphi\) is also hereditary with respect to the above properties for connected closed subsets \(A\) of \(X\). The author applies his result in proving a characterization of cohomological dimension for compacta (resp. shape dimension for continua) generalizing the corresponding results of R. D. Edwards and A. Dranishnikov for cohomological dimension.