Lusternik-Schnirelmann category and the connectivity of \(X\)
The author continues to research a new numerical topological invariant of Lusternik-Schnirelmann type -- the \(\omega\)-weighted cone length \({\text{cl}}^{\omega}(X)\) -- introduced in [\textit{N. Scoville}, JP J. Geom. Topol. 11, No. 3, 209--233 (2011; Zbl 1237.55002)]NEWLINENEWLINETheorem. Let \(a\geq 1\) be an integer and \(q\) a rational number with \(q\geq 1/a\). Then for any \(\omega\) there exists an \(a\)-connected space \(X_q\) with \({\text{cl}}^{\omega}(X)=q\).NEWLINENEWLINEThis theorem is based on the following fact. Recall the definition of a categorical sequence for \(X\), see [\textit{R. Nendorf}, \textit{N. Scoville} and \textit{J. Strom}, Algebr. Geom. Topol. 6, 809--838 (2006; Zbl 1131.55001)].NEWLINENEWLINECorollary. Let \(X\) be a space with \({\text{cat}} X=n\) and let \(\sigma_X=\{m_1, \ldots, m_n\}\) be a categorical sequence. Then \(a+\displaystyle \sum_{k=1}^n 1/(m_k-1)\leq {\text{cl}}^{\omega}(X)\) where \(a=1\) for \(m_1=1\) and \(a=0\) otherwise. Applications to several spaces are presented.
scientific article; zbMATH DE number 6035483
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