论文标题
通过明确的封闭submanifold的同源类实现同源类，这是汤姆（Thom
Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds-
论文作者
论文摘要
这是汤姆（Thom）差异拓扑的经典重要问题。对于紧凑型歧管的同源类别，我们能否通过没有边界的封闭的子手机来意识到这一点？如果在系数环为Z_2的条件下，班级的程度较小或等于外部歧管的尺寸的一半或等于外部歧管的一半。如果在系数环是k是歧管的尺寸的条件下，班级的程度较小或等于6或等于6或等于K-2或K-1，则这也是正确的。作为一项特定研究，例如，对于四维封闭的歧管，例如，已经积极研究了给定第二个同源性类别的封闭和连接表面的拓扑（属）。 在本文中，我们考虑以下类似问题。我们能否通过嵌入（内部）紧凑的歧管（内部）的显式封闭歧管的同源类别来实现紧凑型歧管的同源类别？这个问题被认为是以前问题的变体。我们在可区分图的奇异理论中通过重要理论提出了一个肯定的答案：将给定的平滑图提起到嵌入或获得嵌入的嵌入，使得与规范投影的组成是给定的映射。提出平稳地图和相关基本命题的应用也是本文的主要目的。
It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is Z_2. If the degree of the class is smaller or equal to 6 or equal to k-2 or k-1 under the condition that the coefficient ring is the integer ring where k is the dimension of the manifold, then this is also true. As a specific study, for 4-dimensional closed manifolds, the topologies (genera) of closed and connected surfaces realizing given 2nd homology classes have been actively studied, for example. In the present paper, we consider the following similar problem; can we realize a homology class of a compact manifold by a homology class of an explicit closed manifold embedded in the (interior of the) given compact manifold? This problem is considered as a variant of previous problems. We present an affirmative answer via important theory in the singularity theory of differentiable maps: lifting a given smooth map to an embedding or obtaining an embedding such that the composition of this with the canonical projection is the given map. Presenting this application of lifting smooth maps and related fundamental propositions is also a main purpose of the present paper.
