Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal
Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal
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Seifert 纤维空间的不可约 Heegaard 分裂要么是垂直的，要么是水平的
DOI：
10.1016/s0040-9383(97)00072-4
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
J. Schultens
中科院分区：
文献类型：
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作者：
Y. Moriah;J. Schultens
不可约3-流形分为哈肯流形和非哈肯流形。人们对哈肯流形了解很多，并且这些知识是通过利用它们包含不可压缩表面的事实而获得的。另一方面，人们对非哈肯流形知之甚少。由于我们无法利用不可压缩表面，我们被迫考虑其他方法来研究这些流形。例如，利用 Heegaard 分裂的结构。 Casson 和 Gordon [6] 的结果增强了这种方法，即不可约 Heegaard 分裂要么是强不可约的（参见定义 1.2），要么流形是 Haken。因此，研究Heegaard分裂作为理解3-流形的一种手段，无论它们是否是哈肯流形，都具有新的意义。设M是一个可定向的Seifert纤维空间，具有m个异常纤维和一个可定向的属g基空间。已知这些流形具有“垂直”（参见定义 2.1）Heegaard 分裂 2g# m！ 1. 这些 Heegaard 分裂由 Lustig 和 Moriah 在 [12] 和 [25] 中分类，除非 g" 0 和 0 ( m) 4. 此类中属 2 流形（即 g" 0 和 m" 3）的 Heegaard 分裂由 Boileau 等人[1] 分类，Moriah [14] 使用 Boileau 和 Otal 在 [3] 中的工作分别分类。在这种情况下，有是具有“水平”Heegaard 分裂的流形（参见定义 3.1），Schultens [17] 将流形的 Heegaard 分裂分类为（可定向表面）；并且最近，她证明了 [20] 具有非空边界的可定向 Seifert 纤维空间的所有不可约 Heegaard 分裂都是垂直的。 [24] 对透镜空间的 S、Bonahon 和 Otal [5] 进行了 Heegaard 分裂分类，而 Boileau 和 Otal [2] 对 1 也进行了分类。
Irreducible 3-manifolds are divided into Haken manifolds and non-Haken manifolds. Much is known about the Haken manifolds and this knowledge has been obtained by using the fact that they contain incompressible surfaces. On the other hand, little is known about non-Haken manifolds. As we cannot make use of incompressible surfaces we are forced to consider other methods for studying these manifolds. For example, exploiting the structure of their Heegaard splittings. This approach is enhanced by the result of Casson and Gordon [6] that irreducible Heegaard splittings are either strongly irreducible (see Definition 1.2) or the manifold is Haken. Hence, the study of Heegaard splittings as a mean of understanding 3-manifolds, whether they are Haken or not, takes on a new significance.Let M be an orientable Seifert fibered space with m exceptional fibres and an orientable base space of genus g. These manifolds were known to have ‘‘vertical’’(see Definition 2.1) Heegaard splittings of genus 2g# m! 1. These Heegaard splittings were classified by Lustig and Moriah in [12] and [25], unless g" 0 and 0 ( m) 4. Heegaard splittings of manifolds of genus 2 (ie, g" 0 and m" 3) in this class were classified by Boileau et al.[1] and separately by Moriah [14] using the work of Boileau and Otal in [3]. In this case there are manifolds which have ‘‘horizontal’’Heegaard splittings (see Definition 3.1). Schultens [17] classified Heegaard splittings of manifolds which are (orientable surfaces); S and showed that these are all vertical. More recently, she showed [20] that all irreducible Heegaard splittings of orientable Seifert fibered spaces over an orientable base space with nonempty boundary are vertical. It should be mentioned that Waldhausen [24] classified Heegaard splittings for S, Bonahon and Otal [5] for Lens spaces and Boileau and Otal [2] did so for ¹.