Lusternik-Schnirelmann type invariants for Menger manifolds (Q5949459)

There is a strong analogy between Hilbert cube manifold theory and Menger manifold theory. Using this, the author proves for Menger manifolds results analogous to those proved by \textit{L. Montejano} [Topology Appl. 27, No. 1, 29-35 (1987; Zbl 0646.58013)] for compact and by \textit{R. Y. Wong} [Proc. Am. Math. Soc. 102, No. 3, 720-722 (1988; Zbl 0642.58018)] for noncompact Hilbert cube manifolds.NEWLINENEWLINENEWLINELet \(\mu^n\) be the \(n\)-dimensional universal Menger compactum, and \(\lambda^n=\mu^n\smallsetminus \{\text{point}\}\). The \(n\)-dimensional category of a space \(X\), \(\text{cat}_n(X)\), is the minimum \(k\) for which there exists an open cover \(\{U_i; i=1,\dots,k\}\) of \(X\) such that each inclusion \(U_i\rightarrow X\) is \(n\)-homotopic to a constant map. For an \(n\)-dimensional Menger manifold \(M\), let its \`\` geometric category'', \(\text{gcat}_{\lambda^n}(M)\), be the minimum \(k\) for which there exists an open cover \(\{U_i; i=1,\dots,k\}\) of \(M\) such that each \(U_i\) is homeomorphic to \(\lambda^n\). The author proves that if \(M\) is an \(n\)-dimensional connected compact Menger manifold, then \(\text{gcat}_{\lambda^n}(M)-1=\text{cat}_{n-1}(M)= \text{gcat}_{\lambda^n}(\text{Ker}_{n-1}(M))\), where \(\text{Ker}_{n-1}(M)\) is the \((n-1)\)-homotopy kernel of \(M\) in the sense of Chigogidze.