Degrees of Self-Maps of Simply Connected Manifolds

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 gained importance in the theory of functorial semi-norms on homology. Although inflexibility is expected to 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 rational homotopy types of 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 generalize 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 that reverses orientation. (For this we consider the lack of orientation reversal in the strongest sense possible, i.e., we prove the nonexistence of any self-map of arbitrary negative mapping degree.) Moreover, the manifolds neither split as nontrivial Cartesian products nor as nontrivial connected sums.