1. Homologie effective
We prove that the topological φ-category of a pair (M,N) of topological manifolds is infinite if the algebraic φ-category of the pair of fundamental groups (π₁(M),π₁(N)) is infinite. Some immediate consequences of this fact are also pointed out.
For a compact connected semisimple Lie group we shall prove two results (both related with Singhof’s paper [13]) on the Lusternik-Schnirelmann category of the adjoint orbits of , respectively the 1-dimensional relative category of a maximal torus in . The techniques will be classical, but we shall also apply some basic results concerning the so-called -category (cf. [14]).