Rational Lusternik-Schnirelmann category of fibrations
The (normalized) LS-category of a space \(X\) is the least integer \(n\) such that the Ganea fibration \(p_n: G_n(X)\to X\) admits a section. In the case of a fibration \(F\to E\to B\), the category of the total space \(E\) is bounded by \(\text{cat}(E)\leq \text{cat}(F)+ \text{cat}(B)(\text{cat}(F)+ 1)\). In the paper under review, the authors improve this upper bound in the case of rational spaces by introducing a new invariant called the rational retraction index of \(X\) and denoted by \(\eta_0(X)\). They prove \(\text{cat}(E)\leq \text{cat}_0(F)+ \text{cat}_0(B)(\text{cat}_0(F)+ 2- \eta_0(F))\). The interest of this new result lies in the fact that \(\eta_0(X)\) is often larger than 1.
scientific article; zbMATH DE number 1839051
scientific article