The derived module of a homomorphism
The derived module of a homomorphism
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同态的导出模
DOI：
10.1016/0001-8708(71)90016-8
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发表时间：
1971
影响因子：
1.7
通讯作者：
R. H. Crowell
中科院分区：
文献类型：
--
作者：
R. H. Crowell
纽结理论作为一门数学学科的重要性主要是由于它与其他数学分支的交叉。其中值得注意的是无限离散非阿贝尔群的主题，主题的同源性群体和理论的诺特模块，并研究覆盖空间。一个结构，这是基本的纽结理论，这是讨论的长度在参考文献。[4]是亚历山大矩阵。这个矩阵导致了相应模的定义，有时称为亚历山大模，并由此导致了在参考文献[1]中研究的链模序列。[3]和[5]。尽管在文献中的参考资料，无论是亚历山大矩阵和模块序列之间的连接，也不是基本的几何意义，这些想法是现成的或熟悉的开始学生的纽结理论。本文的目的是提供一个良好的基础。在这种方法的基本定义，即导模的同态，是简单而优雅的。它在计算上是有用的，正如我们在上一节中所展示的，因为它在概念上非常接近福克斯基于他的自由微分[7]对群表示的亚历山大矩阵的定义。它导致很好的代数处理群和模序列的参考文献。[2]和[9]中所述。此外，它很快就意味着重要的几何描述方面的同源性覆盖空间，讨论，例如，米尔诺在参考。[ll]。
The importance of knot theory as a mathematical discipline is due primarily to its intersection with other branches of mathematics. Notable among these are the subject of infinite discrete non-Abelian groups, topics in the homology of groups and the theory of Noetherian modules, and the study of covering spaces. One construction, which is basic to knot theory and which is discussed at length in Ref.[4], is that of the Alexander matrix. This matrix has led to the definition of a corresponding module, sometimes called the Alexander module, and thence to the link module sequence studied in Refs.[3] and [5]. In spite of references in the literature, neither the connection between the Alexander matrix and the module sequence nor the fundamental geometric significance of these ideas is readily available or familiar to beginning students of knot theory. The purpose of this paper is to provide a good foundation. In this approach the basic definition, that of the derived module of a homomorphism, is simple and elegant. It is computationally useful, as we show in the last section, since it is conceptually very close to the definition by Fox of the Alexander matrix of a group presentation based on his free differential calculus [7]. It leads nicely to the algebraic treatments of group and module sequences in Refs.[2] and [9]. Moreover, it quickly implies the important geometric description in terms of the homology of covering spaces, as discussed, for example, by Milnor in Ref.[ll].