An algorithm to decide if a 3-manifold is a Haken manifold
An algorithm to decide if a 3-manifold is a Haken manifold
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判断 3 流形是否为 Haken 流形的算法
DOI：
10.1016/0040-9383(84)90039-9
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发表时间：
1984
期刊：
影响因子：
--
通讯作者：
U. Oertel
中科院分区：
文献类型：
--
作者：
W. Jaco;U. Oertel
紧的3-流形M称为Haken流形，如果它包含一个不同于S 2的适当嵌入的内射曲面F(如果n~(F)-，NL(M)是内射的，则曲面F_c_M是内射的)。近些年来，三维拓扑学的大部分工作都集中在对不可约(M中的每个2-球界为M中的一个3-胞格)、Haken流形的理解上。3-流形是由某种有限表示给出的，例如三角剖分或句柄分解，并根据表示定义一组法线曲面。从本质上讲，这种研究三维流形中曲面的方法已经出现在克尼瑟的工作中[5]。结果表明，无论给出什么表示，流形中的闭内射曲面都会出现在流形的法线曲面之间(直到保纯)。根据Haken发展的理论，给定流形的表示，在法线曲面之间存在有限数量的“基本曲面”，从这些“基本曲面”可以构造所有的法线曲面。因此，如果流形中存在不同于S 2的内射曲面，则它属于可构造曲面。自那以后，哈肯开发了一种算法来确定这样的法线曲面是否是内射的。如果M包含内射曲面，通过逐个检查法线曲面的内射性，Haken算法最终将产生内射法线曲面。然而，这不是一个判定M是否包含内射曲面的有限算法。困难在于“最终”这个词。如果在许多步骤之后，算法没有生成内射曲面，它应该继续还是停止？
A COMPACT 3-manifold M is called a Haken manifold if it contains a properly embedded, injective surface F different from S 2 (the surface F c M is injective if n~(F)--, nl (M) is an injection). Most of the work in 3-dimensional topology in recent years has been in developing an understanding of irreducible (every 2-sphere in M bounds a 3-cell in M), Haken manifolds.In the early 1960s, Haken developed the theory of normal surfaces. A 3-manifold is given by some finite presentation, say a triangulation or a handle-decomposition, and a set of normal surfaces is defined in terms of the presentation. Essentially, this method of studying surfaces in 3-manifolds had already appeared in the work of Kneser [5]. It turns out that a closed, injective surface in the manifold appears (up to isotopy) among the normal surfaces of the manifold, no matter what presentation is given. According to the theory as Haken developed it, given the presentation of the manifold, there exist among the normal surfaces a finite number of" fundamental surfaces" from which all normal surfaces can be constructed. So, if an injective surface, different from S 2, exists in the manifold, then it is among the constructible surfaces. Haken has since developed an algorithm to decide if such a normal surface is injective. If M contains an injective surface, by checking normal surfaces for injectivity one by one, the Haken algorithm will eventually produce an injective normal surface. This is not, however, a finite algorithm for deciding whether M contains an injective surface. The difficulty lies in the word" eventually". If after many steps the algorithm has produced no injective surfaces, should it continue or stop?