Heegaard structures of negatively curved 3-manifolds
Heegaard structures of negatively curved 3-manifolds
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负曲三流形的 Heegaard 结构
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
J. Rubinstein
中科院分区：
文献类型：
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作者：
Y. Moriah;J. Rubinstein
设M是有限体积的可定向完备双曲3-流形，具有d阶尖点ci，…，c^.用M表示在M上通过Dehn手术获得的所有流形的集合，即，如果q ∈ Q，q =（gi，.，qd），设Mq表示第i次qi运算得到的流形。cusp，i = 1，...，d，关于一些预先选择的基础的尖点同调。注意，拓扑上M包含在Mq中，对于所有q。通过M的Heegaard分裂，我们意味着M分解为两个压缩体，如[CG]中所示。Thurston和Jorgenson（见[Thl]）证明了具有有限体积的完备双曲流形的集合是良序集。极限点是具有尖点的流形M，并且流形Mq在Gromov意义下收敛到M，作为max{|PJ|, |RJ|} -> oo对于所有i = 1，.，d，其中qi = Pi/r*。我们的主要结果表明，有界体积的完备双曲3-流形族的Heegaard结构强烈地反映了这种良序性质。我们将证明流形Mg的一致有界亏格的一般不可约Heegaard分裂，其中max{|Pi|, |RJ|}对于alii = 1，.，d，来自M的不可约Heegaard分裂。这可以解释为，Mq的不可约Heegaard分裂收敛于M的这样的分裂，因为Mq收敛于M。换句话说，Heegaard结构和这些流形的许多其他性质一样是对称的“刚性”。在这一段中，我们定义了我们的主要定理所需的一些术语：流形M有一个带有一个0-句柄的句柄分解。令go为M的这种分解中1-柄的最小数目。请注意，通过M上的Dehn手术获得的所有流形都具有Heegaard分裂
Let M be an orientable complete hyperbolic 3-manifold of finite volume and with d ordered cusps ci,... , c^. Denote by M the collection of all manifolds obtained by Dehn surgery on M i.e., if q £ Q, q = (gi,... , qd), let Mq denote the manifold obtained by qi surgery on the i-th. cusp, i = 1,... ,d, with respect to some prechosen basis for the cuspital homology. Note that topologically M is contained in Mq for all q. By a Heegaard splitting for M we mean a decomposition of M into two compression bodies as in [CG]. Thurston and Jorgenson (see [Thl]) proved that the collection of complete hyperbolic manifolds with finite volume is a well ordered set. The limit points are the manifolds M with cusps and the manifolds Mq converge to M in the sense of Gromov as max{|pj|, |rj|} —> oo for all i = 1,... ,d, where qi = Pi/r*. Our main result shows that the Heegaard structure of the family of complete hyperbolic 3-manifolds of bounded volume strongly reflects this well-ordering property. We will show that generically, irreducible Heegaard splittings of uniformly bounded genus of the manifolds Mg, with max{|pi|, |rj|} sufficiently large for alii = 1,... ,d, come from irreducible Heegaard splittings of M. This can be paraphrased as saying that irreducible Heegaard splittings of Mq converge to such splittings of M as Mq converge to M. In other words the Heegaard structure is asymtotically "rigid" as are many other properties of these manifolds. In this paragraph we define some terminology needed for our main theorem: Manifolds M have a handle decomposition with one 0-handle. Let go be the smallest number of 1-handles in such a decomposition of M. Notice that all manifolds obtained by Dehn surgery on M have Heegaard splittings
影响因子：
0.7
作者：
.Ilker S. Yuce-Ilker-S.-Yuce-102800584
通讯作者：
.Ilker S. Yuce-Ilker-S.-Yuce-102800584