The proper Lusternik-Schnirelmann category of semistable one-ended 3-manifolds
Let \(X\) be a topological space. The proper Lusternik--Schnirelmann category of \(X\), denoted \(\mathrm{p-cat}(X)\), is the least integer \(n\) such that \(X\) admits a categorical cover \(\{U_j\}_{j=1}^n\) with the property that for every \(j\) the inclusion of the closure of \(U_j\) \(i_j: \overline{U_j}\to X\) factors up to proper homotopy through \(\mathbb{R}^+\) or \(\{x_0\}\). The study of \(\mathrm{p-cat}(M)\), where \(M\) is a \(3\)-manifold with additional properties, is the main focus of the paper under review. The authors use a Ganea fibration definition of proper category to prove the following conjecture from [\textit{M. Cárdenas}, \textit{F. F. Lasheras} and \textit{A. Quintero}, Math. Proc. Camb. Philos. Soc. 152, No. 2, 223--249 (2012; Zbl 1245.55002)]: If \(M\) is an orientable one-ended semistable open \(3\)-manifold, then \(\mathrm{p-cat}(M)\leq 3\) if and only if \(\pi_1(M)\) is free. Part of the proof utilizes the \(2\)-dimensional Berstein-Svarc theorem due to \textit{A. Dranishnikov} and \textit{Y. Rudyak} [Math. Proc. Camb. Philos. Soc. 146, No. 2, 407--413 (2009; Zbl 1171.55002)].
scientific article; zbMATH DE number 6150727