Smooth spheres in ℝ4 with four critical points are standard
Smooth spheres in ℝ4 with four critical points are standard
复制标题
ℝ4 中具有四个临界点的光滑球体是标准的
DOI：
10.1007/bf01388659
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发表时间：
1985
影响因子：
3.1
通讯作者：
M. Scharlemann
中科院分区：
文献类型：
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作者：
M. Scharlemann
令 M 为 1t 4 中平滑嵌入的 2-球体，其上有一些投影 114__。 11 有四个非简并临界点。这里我们证明 M 是 114 中标准 2 球体的同位素。这解决了 Kuiper [Ku] 提出的问题。该证明基于细川[Ho]首先提出的定理，但存在重大错误（第253页1.3）。该定理也对 [Ki] 的问题 1.1 和 1.2 A 给出了肯定的答案，因此对 1.3 给出了否定的答案。 1.1 的原始解决方案由 Lickorish 提出，使用了 Howie 或 Thurston 和 Gerstenhaber-Rothaus 的深层结果。相比之下，这里的证明是基本的，因为论证是完全独立的和组合的。事实上，这个论点早在 50 年前就已存在。这是一个大纲：w 1 包含主要定理的陈述，以及对其后果的讨论。 w 概述了初步概念和定义。 w 3 是一个关于不等式的简短引理。 w 4 包含证明的主要拓扑论证。 w 5 包含主要组合元素。 w 通过精心构造完成了证明。尽管组合数学令人望而生畏，但它也很美丽。持续使用图论的主要工具（电路、循环、源和汇）几乎不可避免地会导致定理的证明。只有半循环的引入似乎是临时的，但这也是受到 w 3 不等式引理的充分推动。我感谢 N. Kuiper 提供了该引理的更清晰的陈述和证明。 w 1. 主要定理及应用 用两个 2 圆盘 D~ 和 Dh（一纵一横）将 3 球分成四个象限。用罗盘的东北、西北、西南、东南点标记象限。令 N 为通过将两个 1 手柄连接到 3 球而获得的 3 流形，一个手柄连接 SE 到 NE，另一个连接 SW 到 NW。想象一下
Let M be a smoothly imbedded 2-sphere in 1t 4 on which some projection 114__. 11 has four non-degenerate critical points. Here we show that M is isotopic to the standard 2-sphere in 114. This solves a question asked by Kuiper [Ku]. The proof is based on a theorem first claimed by Hosokawa [Ho], but with a major error (p. 253 1.3). This theorem also gives affirmative answers to questions 1.1 and 1.2 A of [Ki], hence a negative answer to 1.3. The original solution to 1.1, proposed by Lickorish, uses deep results of Howie or Thurston and Gerstenhaber-Rothaus. In contrast the proof here is elementary in the sense that the argument is entirely self-contained and combinatorial. Indeed, the argument was available 50 years ago. Here is an outline: w 1 contains the statement of the main theorem, together with a discussion of it consequences. w outlines preliminary notions and definitions. w 3 is a brief lemma on inequalities. w 4 contains the main topological arguments of the proof. w 5 contains the main combinatorial elements. w completes the proof with an elaborate construction. Though the combinatorics is daunting, it is also beautful. Persistent use of the main tools of graph theory (circuits, cycles, sources and sinks) leads almost inevitably to the proof of the theorem. Only the introduction of semi-cycles seems ad hoc, but this too is soundly motivated by the lemma on inequalities of w 3. I am indebted to N. Kuiper for providing a clearer statement and proof of that lemma. w 1. The main theorem and applicationsDivide a 3-ball into four quadrants by two 2-disks, D~ and Dh, one vertical and one horizontal. Label the quadrants by the points of the compass NE, NW, SW, SE. Let N be the 3-manifold obtained by attaching two 1-handles to the 3-ball, one connecting SE to NE, the other connecting SW to NW. Picture the