Constructing framed 4-manifolds with given almost framed boundaries
Constructing framed 4-manifolds with given almost framed boundaries
复制标题
使用给定的几乎框架边界构建框架 4 流形
DOI：
10.1090/s0002-9947-1979-0539917-x
复制
发表时间：
1979
影响因子：
1.3
通讯作者：
S. J. Kaplan
中科院分区：
文献类型：
--
作者：
S. J. Kaplan
给出了两种构造具有给定几乎框架边界的框架4-流形的方法。主要工具是Kirby的框架链接演算的“移动”。给出了纽结的u-不变量的一种新的描述，并将这种描述用于几乎框架三维流形的研究。0.导论.本文的目的是研究给定闭的定向3-流形有界的框架4-流形的类型，并提供构造具有给定边界的框架4-流形的方法。有六个部分。?? 1和2是关于植物的。的主要结果？3是Milnor [8]定理3.1（Milnor）的构造性证明。每一个闭的几乎框架3-流形都有紧的框架4-流形的边界。我们的观点是把给定的3-流形看作是一个具有零柄和所有其它指标为2的柄的可积体的边界。所构造的框架流形可以明确地描绘为这种类型的双曲体。我们的主要工具是Kirby的框架链接演算的两个“移动”[6]。由于证明涉及几个选择，因此可以生成许多具有特定边界的不同框架4-流形。为了说明定理3.1的证明的构造性方面，我们展示了如何找到一个显式的拟体表示，没有指标1或3的句柄，为一个封闭的几乎可平行的4-流形，指标= 16和第二betti数= 22（定理3.3）。这个4-流形同伦等价于库默曲面[3]，这是拓扑学家和代数几何学家都感兴趣的一个例子（见注3.4）。的主要结果？4（定理4.2）是求具有给定边界的框架4-流形的第二种方法。在一般情况下，这些4-流形有较小的指数和第二贝蒂数比那些构造？3，但更难明确生产。提供的公式允许编辑于1976年4月29日收到，并以修订的形式于1977年2月11日和1978年2月16日收到。AMS（MOS）主题分类（1970年）。小学57 A15;中学55 A25、57 A10。
Two methods are presented for constructing framed 4-manifolds with given almost framed boundaries. The main tools are the "moves" of Kirby's calculus of framed links. A new description is given for the ,u-invariant of a knot and this description is used to study almost framed 3-manifolds. 0. Introduction. The purpose of this paper is to study the types of framed 4-manifolds that a given closed, oriented 3-manifold bounds and to provide means of constructing framed 4-manifolds with a given boundary. There are six sections. ?? 1 and 2 are concerned with preliminaries. The main result of ?3 is a constructive proof of the following theorem of Milnor [8]: THEOREM 3.1 (MILNOR). Every closed, almost framed 3-manifold bounds a compact framed 4-manifold. Our point of view is to regard the given 3-manifold as the boundary of a handlebody with a zero handle and all other handles of index 2. The framed manifolds constructed can be pictured explicitly as handlebodies of this type. Our main tools are the two "moves" of Kirby's calculus of framed links [6]. As the proof involves several choices, many different framed 4-manifolds with a particular boundary can be generated. In order to illustrate the constructive aspect of the proof of Theorem 3.1, we show how to find an explicit handlebody presentation, with no handles of index 1 or 3, for a closed almost parallelizable 4-manifold with index = 16 and second betti number = 22 (Theorem 3.3). This 4-manifold is homotopy equivalent to the Kummer surface [3], an example of interest to both topologists and algebraic geometers (see Remark 3.4). The principal result of ?4 (Theorem 4.2) is a second method for finding a framed 4-manifold with given boundary. In general, these 4-manifolds have smaller index and second betti number than those constructed in ?3, but are more difficult to produce explicitly. Formulas are provided allowing the Received by the editors April 29, 1976 and, in revised form, February 11, 1977 and February 16, 1978. AMS (MOS) subject classifications (1970). Primary 57A15; Secondary 55A25, 57A10.