  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. We show that from a
certain dimension on there are infinitely many inflexible manifolds in each
dimension. Besides, we prove flexibility 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 to generalise a conjecture of Copeland--Shar
to the "real world".
  Moreover, we then provide examples of simply-connected smooth compact closed
manifolds in each dimension from dimension 70 on which have the following
properties: They do not admit any self-map which reverses orientation. (For
this we consider the lack of orientation reversal in the strongest sense
possible, i.e. we prove the non-existence of any self-map of arbitrary negative
mapping degree.) Moreover, the manifolds neither split as non-trivial Cartesian
products nor as non-trivial connected sums.
