Heegaard surfaces and the distance of amalgamation
Heegaard surfaces and the distance of amalgamation
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DOI：
10.2140/gt.2010.14.1871
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发表时间：
2008-07
影响因子：
2
通讯作者：
Tao Li
中科院分区：
文献类型：
--
作者：
Tao Li
令$ m_1 $和$ m_2 $为定向不可约3 - 具有连接边界的manifolds，并假设$ \ partial m_1 \ cong \ partial m_2 $。让$ m $成为一个封闭的3 - 通过沿边界上的$ m_1 $将$ m_1 $粘贴到$ m_2 $获得的manifold。我们表明，如果同态粘合的同态性足够复杂，那么$ m $并不是同型$ s^3 $，并且所有小型heegaard $ m $ of $ m $在某种意义上都是标准的。特别是，$ g（m）= g（m_1）+g（m_2）-g（\ partial m_i）$，其中$ g（m）$表示$ m $的Heegaard属。对于具有多个边界组件的某些歧管，该定理也是正确的。
Let $M_1$ and $M_2$ be orientable irreducible 3--manifolds with connected boundary and suppose $\partial M_1\cong\partial M_2$. Let $M$ be a closed 3--manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S^3$ and all small-genus Heegaard splittings of $M$ are standard in a certain sense. In particular, $g(M)=g(M_1)+g(M_2)-g(\partial M_i)$, where $g(M)$ denotes the Heegaard genus of $M$. This theorem is also true for certain manifolds with multiple boundary components.