On genus 2 Heegaard diagrams for the 3-sphere
On genus 2 Heegaard diagrams for the 3-sphere
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关于 3 球面的属 2 Heegaard 图
DOI：
10.1090/s0002-9947-1983-0688963-5
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发表时间：
1983
期刊：
影响因子：
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通讯作者：
T. Kaneto
中科院分区：
文献类型：
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作者：
T. Kaneto
设D是任意亏格为2的Heegaard图，且（a1，a2;？）I，r2）是与D相关联的循环缩减表示。我们要展示吗？I contains？2或Pi'作为循环意义上的子字，如果{（P，？2}#4 {a-'，a '}成立，并且，使用该性质，（a1，a 2; r1，r2）可以变换为平凡的（al，a2; a“，a 1）。通过亏格为2的Poincare猜想的最近正解，我们的结果暗示了判定问题的纯代数算法解：给定的亏格为2的Heegaard分裂的3-流形是否单连通，等价地，是否同胚于3-球面. 1.导论.这是（5）的继续工作，与Homma和Ochiai的实验发现有关（参见。（5，4）），它表明了存在一种优雅而实用的算法的可能性，该算法通过相互替换（定义1）来简化与3-球面S3的亏格2 Heegaard图相关的基本群的表示。它类似于欧几里德算法应用于相对素数。在本文中，我们将以完全的方式建立它（定理2）。Thurston，Bass，Shalen，Meeks-Yau，Gordon-Litherland等人最近公布的结果暗示了Poincare猜想在Heegaard亏格为2时的正解。然后我们的结果意味着解决同构问题的所有表示与亏格2 Heegaard图中的平凡群。换句话说，它给出了一个简单的算法来判定一个亏格为2的Heegaard分裂的三维流形是否同胚于三维球面。为了证明定理2，我们将给出一个关键定理（定理1），它保证了S3的某些Heegaard图所实现的替换的存在性。定理1的证明是基于伪Heegaard图的新概念（定义4）、对伪Heegaard图的修改（定义5）和（4）的结果。在下一节中，我们将精确地陈述我们的结果，并在随后的？？三四最后一次？5，我们将给出一些例子，以补充与我们的结果有关的说明。
Let D be any genus 2 Heegaard diagram for the 3-sphere and (a,, a2; ?I, r2) be the cyclically reduced presentation associated with D. We shall show that ?I contains ?2 or Pi' as a subword in cyclic sense if {(P, ?2} #4 {a -', a '} holds, and that, using this property, (a1, a 2; r1, r2) can be transformed to the trivial one (al, a2; a ", a 1). By the recent positive solution of genus 2 Poincare conjec- ture, our result implies the purely algebraic, algorithmic solution to the decision problem; whether a given 3-manifold with a genus 2 Heegaard splitting is simply connected or not, equivalently, is homeomorphic to the 3-sphere or not. 1. Introduction. This is the continued work of (5) related to the experimental discovery due to Homma and Ochiai (cf. (5, 4)) which indicates the possibility of the existence of an elegant and practical algorithm for simplifying the presentations of the fundamental group associated with genus 2 Heegaard diagrams for the 3-sphere S3 by mutual substitutions (Definition 1). It is similar to Euclidean algorithm applied to relatively prime integers. In this paper, we shall establish it in complete manner (Theorem 2). The recent result announced by Thurston, Bass, Shalen, Meeks-Yau, Gordon-Litherland and others implies the positive solution of Poincare conjecture in case of Heegaard genus 2. Then our result implies the solution to the isomorphism problem with respect to the trivial group among all the presentations associated with genus 2 Heegaard diagrams. In other words, it gives a simple algorithm to decide whether a given 3-manifold with a genus 2 Heegaard splitting is homeomorphic to the 3-sphere or not. In order to prove Theorem 2, we shall show a key theorem (Theorem 1) which assures the existence of the substitution realized by some Heegaard diagram for S3. Our proof of Theorem 1 is based on the new concept of fake Heegaard diagrams (Definition 4), the surgery on them (Definition 5) and the result of (4). In the next section, we shall state our results precisely and prove them in the subsequent ??3, 4. In the last ?5, we shall show some examples for supplemental remarks related to our results.