Spaces with finitely generated cohomology
Suppose that a compact metric space \(X\) has the Čech cohomology group \(H^*(X;\mathbb{Z})\) finitely generated. The problem if there is a (not necessarily finite) polyhedron \(P\) and a map \(f:X\to P\) such that \(f^*:H^* (P;\mathbb{Z}) \to H^*(X;\mathbb{Z})\) is an isomorphism is important to investigate the stability theorem in shape theory. Here \(H^*(X;\mathbb{Z})= \sum^\infty_{k=1} H^k (X;\mathbb{Z})\). Partial answers were obtained by \textit{L. Demers} [Fundam. Math. 90, 1-9 (1975; Zbl 0317.54041)] in the case of locally pathwise connected and simply connected spaces and by \textit{R. Geoghegan} and \textit{R. C. Lacher} [ibid. 92, 25-27 (1976; Zbl 0339.55012)] in the case of shape simply connected and finite-dimensional compacta. The author and \textit{J. J. Walsh} [Topology Appl. 40, No. 3, 203-219 (1991; Zbl 0789.55001)] employed Geoghegan and Lacher's result and found the problem's importance for cohomological dimension theory. In this paper the author shows the following answer. If all the Čech cohomology groups of a space \(X\) are finitely generated and \(H^m(X)\) is free for some \(m\geq 2\), then there is a metrizable space \(Y\in C^{m-1}\cap LC^\infty\), and a map \(f:X\to Y\) such that \(f^*:H^k (Y)\to H^k(X)\) is an isomorphism for all \(k\geq m\). Moreover, if \(\dim X< \infty\) or \(X\) is uniformly movable, then we can take an \((m-1)\)-connected finite polyhedron \(P\) as the above \(Y\).
scientific article; zbMATH DE number 1761004