Fibrewise construction applied to Ljusternik-Schnirelman category
The Ljusternik-Schnirelman category, \(\text{cat}(X)\), of a space \(X\) is the least integer \(n\) for which \(S\) can be covered by \(n+1\) open sets contractible in \(X\). The Ljusternik-Schnirelman category is a homotopy type invariant; however, it is not easy to compute. The authors define a variant of the LS-category denoted by \(Qcat\). It is obtained by applying a base point version of the functor \(Q=\Omega^{\infty}\Sigma^{\infty}\) fibrewise to the Ganea fibrations. \textit{Y. B. Rudyak} in [Topology 38, 37-55 (1999; Zbl 0927.55005)] defined an invariant, denoted by \(\sigma cat\), which approximates \(cat\). The authors show that \(\text{cat}(X) \geq Q\text{cat}(X) \geq \sigma \text{cat}(X)\). The advantage of \(Qcat\) is that it approximates \(cat\) better: in the case of a rational space \(X\), \(Q\text{cat}(X)=\text{cat}(X)\) and \(\sigma \text{cat}(X)\) equals the Toomer invariant. Futhermore, the authors show that \(Q\text{cat}(X \times Y) \leq Q\text{cat}(X)+Q\text{cat}(Y)\). As the authors say, the invariant \(Qcat\) measures the failure of the formula \(\text{cat}(X \times S^r)=\text{cat}(X)+1\). For any complex \(X\) with two cells, \(Q\text{cat}(X)<\text{cat}(X)\) if and only if \(\text{cat}(X \times S^r)=\text{cat}(X)\) for some \(r \geq 1\).
scientific article; zbMATH DE number 1850617
scientific article