BRANCHED COVERING SPACES AND THE QUADRATIC FORMS OF LINKS, II
BRANCHED COVERING SPACES AND THE QUADRATIC FORMS OF LINKS, II
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分支覆盖空间和链接的二次形式，II
DOI：
10.2307/1969717
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发表时间：
1954
影响因子：
4.9
通讯作者：
R. Fox
中科院分区：
文献类型：
--
作者：
R. Kyle;R. Fox
定向流形的同调群中的自链接由Seifert [12]引入，并应用于纽结的循环覆盖[13]。Seifert [12]、Burger [2]、Blanchfield和Fox [1]定义了自联不变量的或多或少完整的系统。Puppe [9]、Kneser和Puppe [8]以及凯尔[6]证明了所谓的纽结或链环的二次型是由第二个循环覆盖的同调群中的自连接决定的。这是本文的目的是二元论这一理论。自链接的二元化是在？2;同调群被上同调群所取代，并且通过上同调群中的乘积运算-a在同调群中自连接。更确切地说，在同调群的扭子群中的自链接被上同调群的扭子群中的对偶运算所取代。对偶理论当然更一般，因为它适用于任意复形，而不仅仅是定向流形。它在本质上也更具有代数性。计算可以从一个规则的细胞复杂的关联矩阵，不需要几何确定的交叉。的计算，利用了对角映射的链近似，并在这里特别感兴趣的情况下更详细地表示，即三维流形的情况。文[5]为分支覆盖空间的拓扑理论奠定了坚实的基础，文[5]为分支覆盖空间的拓扑理论奠定了坚实的基础。将[5]的3个结果放入适合于随后计算的形式中。在哪？4.给出了一种驯服环的分支覆盖的方法。在哪？5.证明了Seifert关于第二圈分支覆盖的结果可以由此推出. 686
Self-linking in the homology groups of an oriented manifold was introduced by Seifert [12] and applied to the cyclic coverings of knots [13]. More or less complete systems of invariants of self-linking were defined by Seifert [12], Burger [2] and Blanchfield and Fox [1]. It was shown by Puppe [9], Kneser and Puppe [8], and Kyle [6] that the so-called quadratic form of a knot or link is determined by the self-linking in the homology groups of the second cyclic covering. It is the object of this paper to dualize this theory. The dualization of self-linking is carried out in ? 2; homology groups are replaced by cohomology groups and self-linking in the homology groups by a product operation -a in the cohomology groups. More precisely self-linking in the torsion subgroups of the homology groups is replaced by a dual operation in the torsion subgroups of the cohomology groups. The dualized theory is, of course, more general, in the sense that it applies to arbitrary complexes and not just to oriented manifolds. It is also more algebraic in nature. The calculations can be made from the incidence matrices of a regular cell-complex, and do not require the geometric determination of intersections. Calculation of --, makes use of a chain approximation to the diagonal map and is indicated in more detail for the case of special interest here, the case of a 3-dimensional manifold. The topological theory of branched covering spaces has been put on a solid foundation in [5], and in ? 3 results of [5] are put into a form suitable for the calculations to follow. In ? 4 a method for the calculation of in the cyclic branched coverings of a tame knot or link is given. In ? 5 it is shown that Seifert's results on the second cyclic branched covering can be deduced from this. 686