Algebraic classification of linking pairings on 3-manifolds
Algebraic classification of linking pairings on 3-manifolds
复制标题
3-流形上连接配对的代数分类
DOI：
10.1007/bf01457818
复制
发表时间：
1980
影响因子：
1.4
通讯作者：
Sadayoshi Kojima
中科院分区：
文献类型：
--
作者：
A. Kawauchi;Sadayoshi Kojima
链接被理解为一对（G，~B），使得G是有限阿贝尔群，~B是非奇异的对称双线性对G x G~Q/Z。用一个矩阵来表示~B相对于G的循环分裂的生成元的连接（G，~）是很方便的，除非可能发生混淆。在拓扑学研究中，链接现象经常出现。例如，给定一个闭的定向3-流形M，我们有一个由Poincar对偶定义的唯一连接（~HI（M），q~M），其中“rHI（M）”是整同调群Hi（M）的挠部分.本文的目的是在块和下完全确定所有连通（直到同构）的交换半群9 l的结构，并观察到任何连通都同构于闭连通定向3-流形M的连通~B~。为了做到这一点，我们将提出一个完整的系统的不变量的同构联系，这是自然产生的，从我们的目的。这样一个完整的系统已经知道塞弗特[ 11]的情况下，奇素数群，并在一般情况下由伯格[2，萨茨5]在闵可夫斯基的美丽的理论。虽然它们之间有直接或间接的关系（参见。[5]我们在这里不讨论它们之间的关系。
A linking is understood as a pair (G, ~b) such that G is a finite abelian group and ~b is a nonsingular, symmetric bilinear pairing G x G~Q/Z. It is convenient to identify a linking (G, ~) with a matrix which represents ~b relative to the generators of a cyclic splitting of G unless confusion might occur. The linking occurs often in the study of topology. For example, given a closed oriented 3-manifold M, we have a unique linking (~HI(M),q~M) defined by the Poincar~ duality, where "rHI(M ) is the torsion part of the integral homology group Hi(M). The purpose of this paper is to determine completely the structure of the abelian semigroup 9l of all linkings (up to isomorphism) under block sum and to observe that any linking is isomorphic to the linking ~b~ of a closed connected oriented 3-manifold M. To do this, we shall present a complete system of invariants of isomorphic linkings, which arises naturally from our purpose. Such a complete system had already been known by Seifert [ 11] in the case of odd-primary groups, and in general by Burger [2, Satz 5] in terms of Minkowski's beautiful theory. Though they are related directly or indirectly to each other (cf. Fox [5]), we do not discuss here any relation between them.