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<p>For any pair of integers <i>g</i> and <i>n</i> with <i>g</i> ⩾ 3 and 1 ⩽ <i>n</i> ⩽ <i>g</i>, we build a 3-manifold with a distance-2, genus-<i>g</i> Heegaard splitting so that (1) it contains <i>n</i> pairwise disjoint and nonisotopic essential tori; (2) after it is cut open along these tori, one resulting piece is hyperbolic while the others are small Seifert fibered spaces; (3) it provides a substantial result to the rank versus genus problem. These generalize a result in Qiu and Zou (2019).</p>

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Distance two Heegaard splittings, JSJ decompositions and ranks of 3-manifolds

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Wenjie Diao,
  • Yanqing Zou

摘要

For any pair of integers g and n with g ⩾ 3 and 1 ⩽ n ⩽ g, we build a 3-manifold with a distance-2, genus-g Heegaard splitting so that (1) it contains n pairwise disjoint and nonisotopic essential tori; (2) after it is cut open along these tori, one resulting piece is hyperbolic while the others are small Seifert fibered spaces; (3) it provides a substantial result to the rank versus genus problem. These generalize a result in Qiu and Zou (2019).