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<p>Let <i>V</i> ⋃<sub><i>S</i></sub> <i>W</i> be a Heegaard splitting of <i>M</i> with distance <i>n</i> ≥ 2 and <i>F</i> a boundary component of <i>∂_V</i>. A simple closed curve <i>J</i> in <i>F</i> is called distance degenerating if the distance of <i>M</i><sub><i>J</i></sub> = <i>V</i><sub><i>J</i></sub> ⋃<sub><i>S</i></sub> <i>W</i> is less than <i>n</i>, where <i>M</i><sub><i>J</i></sub> is obtained by attaching a 2-handle to <i>M</i> along <i>J</i>. In this paper, by considering the distance between <i>J</i> and the image of the essential disks of <i>W</i> under the projection map, we obtain a sufficient condition for the diameter of the set of distance degenerating curves in <i>F</i> to be bounded in <i>C</i>(<i>F</i>). Moreover, for <i>F</i> = <i>∂M</i>, an upper bound of the diameter of the set of the boundary reducible curves in <i>F</i> is given under some circumstance.</p>

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A Sufficient Condition for the Set of Distance Degenerating Handle Additions to be Bounded

  • 期刊论文

  • 发表日期：

    2025年11月15日

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  • Liyuan Ma,
  • Fengchun Lei,
  • Xudong Zhang

摘要

Let V ⋃S W be a Heegaard splitting of M with distance n ≥ 2 and F a boundary component of ∂_V. A simple closed curve J in F is called distance degenerating if the distance of MJ = VJ ⋃S W is less than n, where MJ is obtained by attaching a 2-handle to M along J. In this paper, by considering the distance between J and the image of the essential disks of W under the projection map, we obtain a sufficient condition for the diameter of the set of distance degenerating curves in F to be bounded in C(F). Moreover, for F = ∂M, an upper bound of the diameter of the set of the boundary reducible curves in F is given under some circumstance.