Heegaard genus, degree-one maps, and amalgamation of 3-manifolds

Pub Date : 2022-07-07 DOI:10.1112/topo.12253

Tao Li

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Abstract

Let $M = W \cup_{T} V$ be an amalgamation of two compact 3-manifolds along a torus, where $W$ is the exterior of a knot in a homology sphere. Let $N$ be the manifold obtained by replacing $W$ with a solid torus such that the boundary of a Seifert surface in $W$ is a meridian of the solid torus. This means that there is a degree-one map $f : M \to N$ , pinching $W$ into a solid torus while fixing $V$ . We prove that $g (M) ⩾ g (N)$ , where $g (M)$ denotes the Heegaard genus. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.

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高格属，度一映射，和3流形的合并

设M = W∪T V $M=\mathcal {W}\cup _\mathcal {T} \mathcal {V}$是沿环面两个紧致3流形的合并，其中W $\mathcal {W}$是同调球中一个结的外部。设N $N$为用实体环面代替W $\mathcal {W}$得到的流形，使得W $\mathcal {W}$中的Seifert曲面的边界是实体环面的子午线。这意味着存在一个一级映射f: M→N $f\colon M\rightarrow N$，将W $\mathcal {W}$捏成一个实体环面，同时固定V $\mathcal {V}$。我们证明g (M)小于g (N) $g(M)\geqslant g(N)$，其中g (M) $g(M)$表示Heegaard属。一个直接的推论是，卫星结的隧道数至少与其模式结的隧道数一样大。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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