折叠映射以使奇异集的连通分量增加2的外科手术操作
Surgery operations to fold maps to increase connected components of singular sets by two
摘要：
在几何学中，以构造性的方式理解流形的拓扑和可微结构是基础和重要的。这在一般情况下是困难的，特别是对于高维流形。作者对此很感兴趣，并试图通过构造显式折叠映射来理解流形：可微映射局部表示为Morse函数的乘积映射和开球上的恒等映射。在研究流形时，通过观察奇点、值和原像（的集合），折叠映射一直是基本的和有用的，正如Thom和Whitney的开创性研究以及Kobayashi、Saeki、Sakuma等的最近研究所显示的那样。在这里，显式流形上的显式折叠映射的建设是困难的。作者构造了几个显式的fold映射族，并研究了容许这类映射的流形。主要的基本方法是外科手术在Kobayashi和Saeki的研究的启发下，作者于1996年引入了将余维数为负的一般可微映射变形到平面上以保持流形的可微结构的操作等，并在此基础上，我们去掉了一个（浸入）子流形，附加一个新的映射，得到一个新的折叠映射，使得由奇点组成的集合的连通分支数增加.在本文中，我们调查的情况下，数字增加了两个，并获得了一种新的类型的情况下。
英文摘要：
In geometry, understanding the topologies and the differentiable structures of manifolds in constructive ways is fundamental and important. It is in general difficult, especially for higher dimensional manifolds. The author is interested in this and trying to understand manifolds via construction of explicit fold maps: differentiable maps locally represented as product maps of Morse functions and identity maps on open balls. Fold maps have been fundamental and useful in investigating the manifolds by observing (the sets of) singular points and values and preimages as Thom and Whitney's pioneering studies and recent studies of Kobayashi, Saeki, Sakuma, and so on, show. Here, construction of explicit fold maps on explicit manifolds is difficult. The author constructed several explicit families of fold maps and investigated the manifolds admitting the maps. Main fundamental methods are surgery operations (bubbling operations), the author recently introduced motivated by Kobayashi and Saeki's studies such as operations to deform generic differentiable maps whose codimensions are negative into the plane preserving the differentiable structure of the manifold in 1996 and so on. We remove a neighborhood of a (an immersed) submanifold consisting of regular values in the target space, attach a new map and obtain a new fold map such that the number of connected components of the set consisting of singular points increases. In this paper, we investigate cases where the numbers increase by two and obtain cases of a new type.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
