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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Distance degenerating handle additions
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by Liang Liang, Fengchun Lei and Fengling Li PDF
Proc. Amer. Math. Soc. 144 (2016), 423-434 Request permission

Abstract:

Let $M=V\cup _{S}W$ be a Heegaard splitting of a 3-manifold $M$ and let $F$ be a component of $\partial M$ lying in $\partial _{-}V$. A simple closed curve $J$ in $F$ is said to be distance degenerating if the distance of $M_{J}=V_{J}\cup _{S}W$ is less than the distance of $M=V\cup _{S}W$ where $M_{J}$ is the 3-manifold obtained by attaching a 2-handle to $M$ along $J$. In this paper, we will prove that for a strongly irreducible Heegaard splitting $M=V\cup _{S}W$, if $V$ is simple or $M=V\cup _{S}W$ is locally complicated, then the diameter of the set of distance degenerating curves in $F$ is bounded.
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Additional Information
  • Liang Liang
  • Affiliation: School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China
  • Address at time of publication: School of Mathematics, Liaoning Normal University, Dalian 116029, People’s Republic of China
  • Email: liang_liang@aliyun.com
  • Fengchun Lei
  • Affiliation: School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China
  • Email: fclei@dlut.edu.cn
  • Fengling Li
  • Affiliation: School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China
  • MR Author ID: 893090
  • Email: dutlfl@163.com
  • Received by editor(s): June 26, 2014
  • Received by editor(s) in revised form: September 2, 2014, December 7, 2014, and December 8, 2014
  • Published electronically: June 24, 2015
  • Additional Notes: The second author was supported by the Fundamental Research Funds for the Central Universities (No. DUT14ZD208) and partially supported by grant No.11329101 of NSFC
    The third author was supported by the Fundamental Research Funds for the Central Universities (No. DUT14LK12) and partially supported by two grants No.11101058 and No.11329101 of NSFC
  • Communicated by: Martin Scharlemann
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 423-434
  • MSC (2010): Primary 57N10; Secondary 57M50
  • DOI: https://doi.org/10.1090/proc/12688
  • MathSciNet review: 3415608