A loop theorem for duality spaces and fibred ribbon knots
A loop theorem for duality spaces and fibred ribbon knots
复制标题
对偶空间和纤维带结的环定理
DOI：
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发表时间：
1983
期刊：
影响因子：
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通讯作者：
C. Gordon
中科院分区：
文献类型：
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作者：
A. Casson;C. Gordon
本文证明了三维对偶空间边界上曲面的Loop定理的一个版本，三维对偶空间是指一个类似于三维流形的空间，它只在某个无扭系数域上满足Poincar~6-Lefschetz对偶的适当形式。我们的动机来自于这样一个事实，即这样的空间出现的无限循环覆盖的某些4流形中出现的研究结和谐，并作为我们的定理的主要应用，我们表明，如果一个在3-球的一个bridred结是一个丝带结，那么它的monodromy延伸到一个briddbody。我们通过研究曲面的平面覆盖来接近回路定理，如Papakyriakopoulos的原始论文[11]和Maskit的后续工作[9]。W包含了对这些问题的简单几何处理。W的主要结果是，对偶空间实际上满足对偶与（扭曲）系数模的商的基本群环的任何权力的增广理想。在W4中，W167 2和3的结果，连同一个代数引理上的交集的一个群环的增广理想的权力，被用来证明循环定理的3维对偶空间。(For为了读者的方便，一个证明的代数temma包括作为附录。w包含上述纤维丝带结的应用。在W中，W的结果被用于获得关于可收缩4-流形的边界中的结的一些问题的有限量的信息。在w中，我们将我们的方法应用于结一致性的另一个方面，并表明对于一端具有理性各向异性纤维结（参见[7]）的任何一致性，将结的补集包含到一致性的补集中会导致基本群的注入。对于环面结，这个问题是由Scharlemann提出的[14]。
In this paper we prove a version of the loop theorem for surfaces in the boundary of a 3-dimensional duality space, i.e. a space which resembles a 3manifold only in that it satisfies the appropriate form of Poincar6-Lefschetz duality over some field of untwisted coefficients. Our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance, and as the main application of our theorem we show that if a fibred knot in the 3-sphere is a ribbon knot, then its monodromy extends over a handlebody. We approach the loop theorem via the study of planar coverings of a surface, as in the original paper of Papakyriakopoulos [11] and the subsequent work of Maskit [9]. w contains a simple geometric treatment of these matters. The main result of w is that a duality space actually satisfies duality with (twisted) coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal. In w 4, the results of w167 2 and 3, together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring, are used to prove the loop theorem for 3-dimensional duality spaces. (For the reader's convenience a proof of the algebraic temma is included as an appendix.) w contains the application to fibred ribbon knots mentioned above. In w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4-manifolds. In w we apply our methods to another aspect of knot concordance, and show that for any concordance with a rationally anisotropic fibred knot (see [7]) at one end, the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups. For torus knots, this question was raised by Scharlemann [14].