论文标题
手术操作以折叠地图以构建折叠地图，其限制对单数组的限制可能不是嵌入
Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings
论文作者
论文摘要
构建Morse函数及其更高的尺寸版本或折叠图在研究拓扑和可区分歧管的可区分结构方面是通过Morse函数，折叠图和更一般的通用图表来研究的基础，重要且具有挑战性的。它是可区分地图和对流形几何形状应用的奇点理论的重要且有趣的分支之一。在本文中，我们介绍了折叠地图，其中包含其Reeb空间的同种学环的信息。 Reeb空间被定义为所有预段的所有连接组件的空间，在合适的情况下，继承了拓扑信息，例如同源性组和歧管的共同体学环。以前，作者在各种情况下证明了折叠地图的构造：关键方法是手术操作以进行歧管和地图，在本文中，我们提出了更多有用的手术操作，我们构建了新的折叠地图。更准确地说，具有奇异值集的折叠图：平滑地图的奇异值集是所有单数点集的图像，并注意，对于折叠地图，所有单数点的集合都是无边界的封闭子手机的集合，并且对它们的限制是编辑1。
Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions, fold maps and more general generic maps. It is one of important and interesting branches of the singularity theory of differentiable maps and applications to geometry of manifolds. In this paper we present fold maps with information of cohomology rings of their Reeb spaces. Reeb spaces are defined as the spaces of all connected components of all preimages, and in suitable situations inherit topological information such as homology groups and cohomology rings of the manifolds. Previously, the author demonstrated construction of fold maps in various cases : key methods are surgery operations to manifolds and maps and in this paper, we present more useful surgery operations and by them we construct new fold maps. More precisely, fold maps with singular value sets with crossings: the singular value set of a smooth map is the image of the set of all singular points and note that for fold maps, the set of all singular points are closed submanifolds without boundaries and the restrictions to them are immersions of codimension 1.
