Holonomy-nilpotent fibrations and rational Lusternik-Schnirelmann category
An interesting relation between the category of the total space and the category of the fibre for ``almost trivial'' fibrations \(F \overset {J} {} {E} B\) is proved. More precisely, suppose that \(\pi_*(J) \otimes \mathbb{Q}\) is injective and that some primitive element \(h \in H_{\text{even}} (\Omega B;\mathbb{Q})\) acts locally nilpotent in \(H_*(F)\) for the holonomy operation, then \(\text{cat}_0(E) \geq \text{cat}_0(F) + 1\). Here \(\text{cat}_0\) means rational category, i.e. category of the rationalization. A generalization is also given when finitely many linearly independent primitive elements of \(H_{\text{even}} (\Omega B; \mathbb{Q})\) act locally nilpotently on \(H_*(F; \mathbb{Q})\).
rational Lyusternik-Shnirel'man category
scientific article; zbMATH DE number 863704