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Distance degenerating handle additions


Authors: Liang Liang, Fengchun Lei and Fengling Li
Journal: Proc. Amer. Math. Soc. 144 (2016), 423-434
MSC (2010): Primary 57N10; Secondary 57M50
DOI: https://doi.org/10.1090/proc/12688
Published electronically: June 24, 2015
MathSciNet review: 3415608
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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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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
Email: dutlfl@163.com

DOI: https://doi.org/10.1090/proc/12688
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
Article copyright: © Copyright 2015 American Mathematical Society