Tangential thickness of manifolds

A notion of tangential thickness of a manifold is introduced. An extensive calculation within the class of lens and fake lens spaces leads to a classification of such manifolds with thickness 1, 3 or 2k, for k >= 1. On the other hand, calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants.