A measure of the deviation from there being fibrations between a pair of compact manifolds

In this paper we first study the topology of certain manifold complements. The obtained results are then used to show that the φ-category (cf. Lusternik-Schnirelmann category) of some pairs of manifolds is infinite. Finally, in the last section the equivariant case is considered, examples of equivariant mappings with infinitely many critical orbits being provided.