Homology classes which are represented by graph links
Homology classes which are represented by graph links
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由图链接表示的同源类
DOI：
10.1090/s0002-9939-1985-0776213-8
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
Koichi Yano
中科院分区：
文献类型：
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作者：
Koichi Yano
我们给出了图流形的一维同调类由图链接表示的充要条件。本文的目的是给出图流形的一维同调类用图链接表示的充要条件。为此，我们使用 Jaco-Shalen 和 Johannson 的所谓环面分解定理（另见 Waldhausen [2]）定义图流形的 Jaco-Shalen-Johannson 复形 (?1)，主要结果如下 (?2)： 定理。设 M 是 S1 X S2 和 p 的图流形：M -T WM 到 M 的 Jaco-Shalen-Johannson 复形的自然映射。那么当且仅当 Hl(WM; Z) 中 p*o(a) = 0 时，H1(M; Z) 的元素 a 可以用图链接表示。如果环境流形 M 不是 S1 X S2 的素数，则映射 p: M -T WM 的同伦类通常不是唯一的，这使得语句变得复杂 (?3)。这些结果的证明基于[3]中对全局图链接的研究。 1. 预备工作。在本文中，流形是紧致的、有向的、三维的、具有环边界的，并且链接是有向的。如果 M 中存在一族不相交嵌入的环面，使得通过沿这些环面切割 M 获得的流形的每个连通分量是表面上 S1 丛的总空间，则流形 M 是图流形。如果链接的外部是图流形，则该链接称为图链接。如果一个流形的质数分解不包含 S1 X S2，我们就说它与 S1 X S2 是素数。其他术语请参见 Jaco [1] 和 Yano [3]。为了陈述我们的结果，我们需要由 Jaco-Shalen 和 Johannson 提出的以下定理（参见 Jaco [1] 和 Waldhausen [2]）。定理 1.1（雅科·沙伦，约翰逊）。设 M 为哈肯流形，它要么是封闭的，要么是具有不可压缩边界的。然后，存在一个直到环境同位素的唯一 Seifert 子流形 E c M，使得 (1) E 最大，并且 (2) 如果 E' 是不同于 (S3, 0)、(S2 X SI, 0)、(D2 X I, aD 2 X I) 或 (S1 X D2, 0) 的 Seifert 流形对，则每个非简并映射 f: E' -> (M, aM) 与 fo 同伦，这样fo(E') c E。上面的 E 被称为 M 的特征 Seifert 流形。编辑于 1983 年 9 月 19 日收到，修订后的形式于 1984 年 3 月 14 日。1980 年数学学科分类。主要 57M99、57N10。
We give a necessary and sufficient condition for a one-dimensional homology class of a graph manifold to be represented by a graph link. The purpose of this paper is to give a necessary and sufficient condition for a one-dimensional homology class of a graph manifold to be represented by a graph link. For this, we define the Jaco-Shalen-Johannson complex of a graph manifold (?1), using the so-called torus decomposition theorem due to Jaco-Shalen and Johannson (see also Waldhausen [2]), and the main result is stated as follows (?2): THEOREM. Let M be a graph manifold prime to S1 X S2 and p: M -T WM the natural map to the Jaco-Shalen-Johannson complex of M. Then an element a of H1(M; Z) can be represented by a graph link if and only if p*o(a) = 0 in Hl(WM; Z). If the ambient manifold M is not prime to S1 X S2, the homotopy class of the map p: M -T WM is not unique in general, and this makes the statement complicated (?3). The proof of these results is based on the study of global graph links in [3]. 1. Preliminaries. Throughout this paper, manifolds are compact, oriented, of dimension three and with toral boundary, and links are oriented. A manifold M is a graph manifold if there is a family of disjointly embedded tori in M such that each connected component of the manifold obtained by cutting M along these tori is the total space of an S1-bundle over a surface. A link is called a graph link if its exterior is a graph manifold. We say that a manifold is prime to S1 X S2 if its prime decomposition does not contain S1 X S2. See Jaco [1] and Yano [3] for other terminology. To state our result, we need the following theorem due to Jaco-Shalen and Johannson (see Jaco [1] and also Waldhausen [2]). THEOREM 1.1 (JACO SHALEN, JOHANNSON). Let M be a Haken manifold which is either closed or with incompressible boundary. Then there exists a unique Seifert submanifold E c M up to ambient isotopy such that (1) E is maximal, and (2) if E' is a Seifert manifold pair distinct from (S3, 0), (S2 X SI, 0), (D2 X I, aD 2 X I) or (S1 X D2, 0), then every nondegenerate map f: E' -> (M, aM) is homotopic to fo such that fo(E') c E. The E above is called the characteristic Seifert manifold of M. Received by the editors September 19, 1983 and, in revised form, March 14, 1984. 1980 Mathematics Subject Classification. Primary 57M99, 57N10.