Explicit fold maps on 7-dimensional closed and simply-connected spin manifolds of new classes
Naoki Kitazawa
备注：
21 pages, Sections 4--5 added (4: Acknowledgement 5: Appendices), this improves one of main results of arXiv:1911.11274, this is submitted to a refereed journal
Closed (and simply-connected) manifolds whose dimensions are greater than 4 are classified via sophisticated algebraic and abstract theory such as surgery theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed manifolds in such ways. However, the latter work via geometric and constructive ways is not so difficult. The assumption that the dimensions are not high enables us to handle the manifolds via diagrams for example. It is difficult to study higher dimensional manifolds in these ways, although it is natural and important.
In the present paper, we present such studies via {\it fold} maps, which are higher dimensional variants of Morse functions. The author previously constructed fold maps on 7-dimensional closed and simply-connected manifolds satisfying additional conditions on cohomology rings, including so-called {\it exotic} homotopy spheres. This paper concerns fold maps on such manifolds of a wider class.
7-dimensional closed simply-connected and spin manifolds having 2nd integral cohomology classes whose squares are not divisible by 2 and stable fold maps on them
7-dimensional simply-connected spin manifolds whose integral cohomology rings are isomorphic to that of ${\mathbb{C}P}^2 \times S^3$ admit round fold maps
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。
