Homology cobordisms, link concordances, and hyperbolic 3-manifolds
Homology cobordisms, link concordances, and hyperbolic 3-manifolds
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同源共轭、链接索引和双曲 3 流形
DOI：
10.1090/s0002-9947-1983-0697074-4
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发表时间：
1983
影响因子：
1.3
通讯作者：
Robert Myers
中科院分区：
文献类型：
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作者：
Robert Myers
.设A/03和A/，3是紧的定向3-流形。如果3A/3 = 0，则它们是同源共边的（分别是相对同源共边的）。3M3#0），并且存在光滑紧致的定向4-流形W*，使得dW 4 = A/q-A/3（分别为. dW_4 = M_（？）- A/3）U（A/，3X [0，1]）和H_t（M_3;Z）-H_（？）_（？）定理每一个闭的定向3-流形同调共边于一个双曲3-流形。定理每一个边界非空且不含2-球面的紧致定向3-流形与双曲3-流形是相对同调共边的。在三维流形M3中的两个定向链环L0和L1是协调的，如果在M3 X [0，1]中存在一组光滑的、不相交的、定向的环，使得<3A 2 = L0 -L1，其中L1 <$M3 X {1}，i = 0，1。定理在一个紧的、定向的3-流形A/3中，其边界不包含2-球面的每一个链环都与其外表面是双曲的链环协调。推论。S3中的每一个纽结都与一个外部为双曲的纽结协调。
. Let A/03 and A/,3 be compact, oriented 3-manifolds. They are homology cobordant (respectively relative homology cobordant) if 3A/3 = 0 (resp. 3M3 # 0) and there is a smooth, compact oriented 4-manifold W* such that dW4 = A/q — A/3 (resp. dW4 = M¡ - A/3) U (A/,3 X [0,1]) and Ht(M3;Z) - HJWA;Z) are iso-morphisms, ¡ = 0, 1. Theorem. Every closed, oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. Theorem. Every compact, oriented 3-manifold whose boundary is nonempty and contains no 2-spheres is relative homology cobordant to a hyperbolic 3-manifold. Two oriented links L0 and L, in a 3-manifold M3 are concordant if there is a set A2 of smooth, disjoint, oriented annuli in M X [0,1] such that <3A2 = L0 — Lt, where L, Ç M3 X {/}, i = 0, 1. Theorem. Every link in a compact, oriented 3-manifold A/3 whose boundary contains no 2-spheres is concordant to a link whose exterior is hyperbolic. Corollary. Every knot in S3 is concordant to a knot whose exterior is hyperbolic.