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.
Diffeomorphism types of simply connected $3$-dimensional Mori fibre spaces
School of Mathematics, Shandong University（山东大学数学学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院；中国科学院大学）；University of Chinese Academy of Sciences（广西民族大学数学科学学院）；School of Mathematical Sciences, Guangxi Minzu University
Rational homotopy and simply-connected 8-manifolds
引入了拓扑空间的有理同伦不变量P，它是上同调的五次张量。当n > 1时，维数不超过5 n-2的闭（n-1）-连通流形的形式等价于P和克劳利和第二作者arXiv：1505.04184引入的Bianchi-Massey张量的消失。我们还证明了闭单连通自旋8-流形群的上同调为S ^2 x S ^6的r重连通和的元素在挠率之前由P的值确定。
