A geometric proof that boundary links are homotopically trivial
A geometric proof that boundary links are homotopically trivial
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边界链接同伦平凡的几何证明
DOI：
10.1016/0166-8641(88)90023-5
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发表时间：
1988
影响因子：
0.6
通讯作者：
D. Dimovski
中科院分区：
文献类型：
--
作者：
D. Dimovski
0.我们在PL类别中工作。n分量链路L=（ai，...，a，i）是S '在S'中的n个不相交副本的嵌入。边界链是其分量在S '中约束不相交的、紧凑的、有向（塞弗特）曲面的链。链路L=（a1，...，a，..）被称为是与链路L '=（a，.. _，ai），如果存在从L到L '的同伦，该同伦除了在多个点处是同伦，其中一个分量具有横截自相交，如图1所示。平凡链环是其组成部分约束不相交的平面圆盘的链环。我听到T。科克伦问R。Edwards（MSRI，Berkeley，1985）是否可以证明边界链路是平凡链路同伦的使用capped摸索，如在M。特殊基本群的5-s-配边定理的弗里德曼证明。Cochran说，这一结果被K.陈或J.米尔诺，但我找不到它在文献中以这种形式。后来，T。Cochran和裁判的标准论点是本文结果的Milnor/I-不变量的消失意味着一个链接是同伦平凡的，所以，人们表明，所有的Milnor p-不变量消失的边界链接。但是，这第二步可能没有出现在文献中，尽管它并不被认为是一个困难的论点。
0. We work in the PL category. An n-component link L=(a,,..., a,,) is an embedding of n disjoint copies of S’in S’. A boundary link is a link whose components bound disjoint, compact, oriented (Seifert) surfaces in S’. A link L=(a,,..., a,,) is said to be homotopic (usually said fink homotopic) to a link L’=(a:,.. _, ai) if there exists a homotopy from L to L’which is an isotopy except at finitely many points, where one of the components has a transverse self-intersection, as shown in Fig. 1. A trivial link is a link whose components bound disjoint flat disks. I heard T. Cochran asking R. Edwards (MSRI, Berkeley, 1985) whether one could prove that boundary links are homotopic to trivial links by using capped gropes, as are used in M. Freedman’s proof of the 5-s-cobordism theorem for special fundamental groups. Cochran said that this result is proved by K. Chen or J. Milnor, but I could not find it in the literature in this form. Later comments by T. Cochran and the referee are that the standard argument for the result of this paper is that vanishing of the Milnor/I-invariants implies that a link is homotopically trivial, so, one shows that all the Milnor p-invariants vanish for a boundary link. But it is this second step that may not be in the literature, although it is not regarded as a difficult argument.