An oriented compact closed manifold is called inflexible if the set of mapping degrees ranging over all continuous self-maps is finite. Inflexible manifolds have become of importance in the theory of functorial semi-norms on homology. Although inflexibility should be a generic property in large dimensions, not many simply-connected examples are known. In this talk we shall see that, from a certain dimension on, there are infinitely many inflexible manifolds in each dimension. We then shall focus on proving inflexibility for large classes of manifolds and, in particular, as a spin-off, for homogeneous spaces. This is an outcome of a lifting result which also permits us to generalise a conjecture of Copeland and Shar.

Time permitting, we shall illustrate how one may use similar techniques to deal with the existence question of simply-connected manifolds which do not admit any orientation reversing self-maps (in the strongest sense possible).