A SURVEY OF APPLICATIONS OF SURGERY TO KNOT AND LINK THEORY
A SURVEY OF APPLICATIONS OF SURGERY TO KNOT AND LINK THEORY
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外科手术在结和链接理论中的应用调查
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
Kent E. Orry
中科院分区：
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作者：
Jerome Leviney;Kent E. Orry
纽结和链环理论研究一个流形如何嵌入另一个流形。给定一个流形嵌入，我们可以通过移除一个点的邻域并用一个嵌入的圆盘对来改变这个邻域中的嵌入。这样，传统的纽结理论，球面嵌入球面的研究，影响了一般的流形嵌入问题。在一维中，流形嵌入问题是纽结和链环理论。本文试图快速调查的作用，外科手术的发展，结和链接理论。外科手术是最有力的工具之一，在处理的问题"在多大程度上是流形（或流形嵌入）唯一确定的同伦类型？"正如我们将看到的，粗略地说，纽结和链环是由它们在余维≥ 3时的同伦类型（更准确地说，是庞加莱嵌入类型）决定的，而在余维2时要复杂得多。我们继续，主要是从历史的角度来看，介绍了最开创性的早期成果的语言和技术，他们首先被发现。纽结理论的这些结果是外科理论最重要的早期应用之一，并促进了它的发展。我们将强调纽结和链接的领域，只提供一个更一般的余维2嵌入问题的简短讨论。特别是，余维2嵌入理论，从庞加莱嵌入类型分类的角度来看，值得一个期待已久的调查文件。本文件不会填补文献中的这一空白。Cappell和Shaneson在[CS78]中对这个主题做了很好的介绍。这项调查决不是全面的，我们事先对遗漏了许多已经做了大量重要工作的领域表示歉意。例如，我们将省略等变纽结理论的广泛主题。我们也将不包括任何讨论的技术德恩手术已被证明是如此宝贵的研究三个流形和经典的结。此外，我们将不涉及浸入理论的相关主题，也几乎不提奇点理论。我们敦促读者咨询的许多优秀的调查，其中包括早期（1977年之前）发展的余维二结理论更深入。卡梅隆戈登[Gor77]和Kervaire-Weber [KW 77]分别关于低维和高维纽结理论的文章非常出色。手术和嵌入理论的详细讨论可以在Ranicki的书[Ran81]中找到。另一方面，我们不知道任何先前存在的综合调查的最新发展联系理论。
Knot and link theory studies how one manifold embeds in another. Given a manifold embedding, one can alter that embedding in a neighborhood of a point by removing this neighborhood and replacing it with an embedded disk pair. In this way traditional knot theory, the study of embeddings of spheres in spheres, impacts the general manifold embedding problem. In dimension one, the manifold embedding problem is knot and link theory. This article attempts a rapid survey of the role of surgery in the development of knot and link theory. Surgery is one of the most powerful tools in dealing with the question “To what extent are manifolds (or manifold embeddings) uniquely determined by their homotopy type?” As we shall see, roughly speaking, knots and links are determined by their homotopy type (more precisely, Poincare embedding type) in codimension ≥ 3 and are much more complicated in codimension two. We proceed, largely, from an historical perspective, presenting most of seminal early results in the language and techniques in which they were first discovered. These results in knot theory are among the most significant early applications of surgery theory and contributed to its development. We will emphasize knotted and linked spheres, providing only a brief discussion of more general codimension two embedding questions. In particular, the theory of codimension two embedding, from the standpoint of classifying within a Poincare embedding type, deserves a long overdue survey paper. The present paper will not fill this void in the literature. Cappell and Shaneson give an excellent introduction to this subject in [CS78]. By no means is this survey comprehensive, and we apologize in advance for the omission of many areas where considerable and important work has been done. For example, we will omit the extensive subject of equivariant knot theory. We will also not include any discussion of the techniques of Dehn surgery that have proven so valuable in the study of three manifolds and classical knots. Furthermore, we will not touch on the related subject of immersion theory, and barely mention singularity theory. We urge the reader to consult one of the many excellent surveys which have covered the early (before 1977) development of codimension two knot theory in more depth. The articles by Cameron Gordon [Gor77] and Kervaire-Weber [KW77] on, respectively, low-dimensional and high-dimensional knot theory are excellent. A detailed discussion of surgery and embedding theory can be found in Ranicki’s book, [Ran81]. On the other hand, we are not aware of any previously existing comprehensive survey of recent developments in link theory.