Comparing Heegaard splittings of non-haken 3-manifolds
Comparing Heegaard splittings of non-haken 3-manifolds
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比较非哈肯 3 流形的 Heegaard 分裂
DOI：
10.1016/0040-9383(95)00055-0
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
Hyam Rubinstein
中科院分区：
文献类型：
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作者：
Hyam Rubinstein
Cerf理论可以用来比较同一个闭可定向三维流形的两个强不可约Heegaard分裂。任何两个分裂曲面都可以是同位的，使得它们相交于一个非空的曲线集合，每个曲线在两个分裂曲面中都是必不可少的。更一般地说，有有趣的分裂面的同位素，在此期间，这种相交性质被保留。作为例子应用，我们给出了Waldhausen定理的新证明，即S3的Heegaard分裂是标准的，以及Bonahon和Otal定理的新证明，即透镜空间的Heegaard分裂是标准的.我们也给出了不可约非哈肯3-流形的稳定化问题的一个解决方案：如果p <$q是这样一个流形的两个分裂的亏格，那么存在亏格为5 p +8 q − 9的公共稳定化。
Cerf theory can be used to compare two strongly irreducible Heegaard splittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collection of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfaces during which this intersection property is preserved. As sample applications we give new proofs of Waldhausen's theorem that Heegaard splittings of S3are standard, and of Bonahon and Otal's theorem that Heegaard splittings of lens spaces are standard. We also present a solution to the stabilization problem for irreducible non-Haken 3-manifolds: If p ⩽ q are the genera of two splittings of such a manifold, then there is a common stabilization of genus 5p + 8q − 9.