$3$维流形上的圆折叠映射及其整上同调环和有理上同调环
Round fold maps on $3$-dimensional manifolds and their integral and rational cohomology rings
摘要：
折叠映射是光滑映射，在每个奇点它被表示为莫尔斯函数和单位映射的乘积映射。简而言之，圆折叠映射是所有奇点的集合同心嵌入的映射。它们和Morse函数一样，在理解拓扑结构和几何上的可微结构中起着重要的作用。本文研究了在平面上可容纳圆折叠映射的$3$维流形的上同调环，发现圆折叠映射的拓扑型与系数环的差异密切相关.这是我们以前研究的一个明确精确的新研究，证明了一个$3$维闭可定向流形是一个所谓的{ \it graph manifold}，或者是一个通过沿着环面在曲面上粘合所谓的圆丛而得到的流形，当且仅当它允许一个圆折叠映入平面。我们还展示了另一个阐述分类的图流形承认这样的地图，其拓扑型是某一最简单的类。
英文摘要：
Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded concentrically. They are, as Morse functions, important in understanding the topologies and the differentiable structures in geometric ways. In the present paper, we study cohomology rings of $3$-dimensional manifolds admitting round fold maps into the plane and see that difference of the coefficient rings and topological types of round fold maps are closely related. This is an explicit precise new study on our previous study, showing that a $3$-dimensional closed and orientable manifold is a so-called {\it graph manifold}, or a manifold obtained by gluing so-called circle bundles over surfaces along tori, if and only if it admits a round fold map into the plane. We also show another exposition on classifications of graph manifolds admitting such maps whose topological types are of a certain simplest class.
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