Cell-like resolutions in the strongly countable \(\mathbb Z\)-dimensional case
The objective of this paper is to prove the following theorem. Theorem 1.1. Suppose that \(X\) is a nonempty compact metrizable space and \(X_1\subset X_2\subset\cdots\) is a sequence of nonempty closed subspaces such that for each \(k\in\mathbb{N}\), \(\dim_{\mathbb{Z}}X_k\leq k<\infty\). Then there exists a compact metrizable space \(Z\), having closed subspaces \(Z_1\subset Z_2\subset\cdots\), and a surjective cell-like map \(\pi: Z\to X\), such that for each \(k\in\mathbb{N}\), (a) \(\dim_{\mathbb{Z}}X_k\leq k\), (b) \(\pi(Z_k)= X_k\), and (c) \(\pi\mid Z_k\) is a cell-like map. Moreover, there is a sequence \(A_0\subset A_1\subset\cdots\) of closed subspaces of \(Z\) such that for each \(k\), \(Z_k\subset A_k\), \(\dim A_k\leq k\), \(\pi\mid A_k: A_k\to X\) is surjective, and for \(k\in\mathbb{N}\), \(\pi\mid A_k: A_k\to X\) is a \(\text{UV}^{k-1}\)-map. (Here a map \(\pi: Z\to X\) is called cell-like if each of its fibers \(\pi^{-1}(x)\) is a cell-like space, and \(\dim_{\mathbb{Z}}\) denotes integral cohomological dimension.) As an application the following nice combinatorial theorem is obtained. Theorem 4.1. Suppose that \((P,\tau_P)\) and \((Q,\tau_Q)\) are compact polyhedra, \(P_0< P\) and \(Q_0< Q\) compact subpolyhedra. Then for every map \(f: P\to Q\), \(f(P_0)\subset Q_0\), there exist a triangulation \(\tau_P'< \tau_P\) and a simplicial map \(f': (P,\tau_P')\to (Q,\tau_Q)\) such that \(\text{dist}(f', f)< 2\cdot\text{mesh}(\tau_Q)\), and \(f'(P_0)\subset Q_0\). The following conjecture is also formulated. Conjecture 1.3. Suppose that \(X\) is a nonempty compact metrizable space and \(X_1\subset X_2\subset\cdots\) is a sequence of nonempty closed subspaces such that for each \(k\in\mathbb{N}\), \(\dim_{\mathbb{Z}}X_k\leq k<\infty\). Then there exists a compact metrizable space \(Z\), having closed subspaces \(Z_1\subset Z_2\subset\cdots\), and a surjective cell-like map \(\pi: Z\to X\), such that for each \(k\in\mathbb{N}\), (a) \(\dim_{\mathbb{Z}}X_k\leq k\), and (b) \(\pi^{-1}(X_k)= Z_k\). Moreover, there is a sequence \(A_0\subset A_1\subset\cdots\) of closed subspaces of \(Z\) such that for each \(k\), \(Z_k\subset A_k\), \(\dim A_k\leq k\), \(\pi\mid A_k: A_k\to X\) is surjective, and for \(k\in\mathbb{N}\), \(\pi\mid A_k: A_k\to X\) is a \(\text{UV}^{k-1}\)-map.
scientific article; zbMATH DE number 2093543