Lens spaces and Dehn surgery
Lens spaces and Dehn surgery
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晶状体间隙和 Dehn 手术
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发表时间：
1989
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通讯作者：
R. Litherland
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作者：
S. Bleiler;R. Litherland
通过卫星结的特征讨论了 Dehn 手术何时产生晶状体间隙的问题。透镜空间 L(2, 1) ，即实投影 3 空间，被证明是无法通过对称结手术获得的。一段时间以来，拓扑学家对何时可以通过对 3 球体中的结进行 Dehn 手术来获得透镜空间的问题感兴趣。众所周知，通过 Dehn 手术对环面结 (Moser [Mo])、某些椒盐卷饼结 (Fintushel-Stern [FS]) 和某些重要的卫星结（实际上是环面结的某些电缆，BaileyRolfsen [BR]）进行手术，可以产生某些晶状体空间。在这篇文章中，我们展示了三流形理论的一些最新发展如何进一步阐明这个问题。在定理 1 中，我们使用 CullerGordon-Luecke-Shalen [CGLS]、Gabai [Ga]、Gordon [Go] 和 Scharlemann [S] 的最新结果来描述如何通过在（非平凡的）卫星结上进行手术来获得晶状体空间。定理1最初是由Wu[Wu]证明的，我们在这里提出一个独立发现的更简洁的证明，它更深入地利用了Gabai和Gordon的临界定理。 Wang [Wa¡] 和 Hempel [H] 也得到了类似的结果。然后我们专门研究何时可以通过 Dehn 手术获得真实的投影 3 空间，即晶状体空间 L(±2,1) 的问题。使用 Thompson [T] 和 Wang [Wa2] 的结果，我们在定理 2 中表明，对非平凡对称结进行手术不会产生该流形。定理 1. 如果对卫星结进行非平凡的 Dehn 手术产生具有循环基本群的流形，则该结是环面结的缆，并且结和手术系数如 [Go，定理 7.5 (iii)，k = 2] 中所示。 IE。结是 (p, q) 圆环结上的 (2pq ± 1,2) 缆线，手术系数为 4pq ± 1，所得流形为 L(4pq ±1,4q )。推论。如果晶状体间距 L 是通过卫星结上的 Dehn 手术获得的，那么
x(L)> 23. 编辑于 1988 年 5 月 20 日收到，并于 1989 年 1 月 23 日修订。 1980 年数学学科分类（1985 年修订版）。主要 57N10、57M25。第一作者得到了 NSF Grant DMS-8602327 的部分支持。第二作者得到了 NSF Grant DMS-8705760 的部分支持。 © 1989 美国数学会 0002-9939/89 每页 1.00 美元+ 0.25 美元
The question of when a lens space arises by Dehn surgery is discussed with a characterization given for satellite knots. The lens space L(2, 1) , i.e. real projective 3-space, is shown to be unobtainable by surgery on a symmetric knot. The problem of when a lens space can be obtained by performing Dehn surgery on a knot in the 3-sphere has been of interest to topologists for some time. It is known that certain lens spaces can arise by Dehn surgery on torus knots (Moser [Mo]), certain pretzel knots (Fintushel-Stern [FS]), and certain nontrivial satellite knots (in fact, certain cables of torus knots, BaileyRolfsen [BR]). In this note we show how some recent developments in 3-manifold theory shed more light on this problem. In Theorem 1 we use recent results of CullerGordon-Luecke-Shalen [CGLS], Gabai [Ga], Gordon [Go], and Scharlemann [S] to characterize how a lens space can be obtained by surgery on a (nontrivial) satellite knot. Theorem 1 was originally proven by Wu [Wu], We present here a somewhat more concise proof, discovered independently, which makes deeper use of the critical theorems of Gabai and Gordon. Similar results have also been obtained by Wang [Wa¡] and Hempel [H]. We then specialize to the question of when real projective 3-space, i.e. the lens space L(±2,1), can be obtained by Dehn surgery. Using results of Thompson [T] and Wang [Wa2], we show in Theorem 2 that no surgery on a nontrivial symmetric knot yields this manifold. Theorem 1. If nontrivial Dehn surgery on a satellite knot yields a manifold with cyclic fundamental group, then the knot is a cable of a torus knot and the knot and surgery coefficient are as in [Go, Theorem 7.5 (iii), k = 2]. I.e. the knot is the (2pq ± 1,2)-cable on a (p, q)-torus knot, the surgery coefficient is 4pq ± 1, and the resulting manifold is L(4pq ±1,4q ). Corollary. If a lens space L is obtained by Dehn surgery on a satellite knot then
x(L)> 23. Received by the editors May 20, 1988 and, in revised form, January 23, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57N10, 57M25. The first author was supported in part by NSF Grant DMS-8602327. The second author was supported in part by NSF Grant DMS-8705760. © 1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page