Lower bounds for the relative Lusternik-Schnirelmann category
Given a pair \((X,A)\) of topological spaces, let \(\text{cat}(X,A)\) denote the relative Lyusternik-Shnirel'man category of \((X,A)\) and set \(\text{cat}(X,X\times S^p)=\text{cat} (X, X\times D^{p+1})\) where \(S^p\) and \(D^{p+1}\) denote the sphere and disk of dimensions \(p\) and \(p+1\), respectively. The author proves that \(\sigma{p+1}\text{cat} (X)\leq \text{cat}(X,X\times S^p)\) and that \(\sigma^{p+2}\text{cat}(X) +1\leq\sigma^1\text{cat}(X, X\times S^p)\leq\sigma ^{p+1}\text{cat}(x)\) where \(\sigma^p\) is the \(\sigma\)-category introduced by \textit{L. Vandembrouck} [Suspension of Ganea fibrations and a Hopf invariant, Topology Appl. 105, No 2, 187-200 (2000)]. The author also notes that \(e(X, X\times S^p)=e(X)+1\) where \(e\) denotes the Toomer invariant.
scientific article; zbMATH DE number 1463443