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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Reducible and $\partial$-reducible handle additions


Authors: Ruifeng Qiu and Mingxing Zhang
Journal: Trans. Amer. Math. Soc. 361 (2009), 1867-1884
MSC (2000): Primary 57M50
DOI: https://doi.org/10.1090/S0002-9947-08-04761-2
Published electronically: November 24, 2008
MathSciNet review: 2465821
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Abstract: Let $M$ be a simple 3-manifold with $F$ a component of $\partial M$ of genus at least two. For a slope $\alpha$ on $F$, we denote by $M(\alpha )$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $\alpha$ on $F$. Suppose that $\alpha$ and $\beta$ are two separating slopes on $F$ such that $M(\alpha )$ and $M(\beta )$ are reducible. Then the distance between $\alpha$ and $\beta$ is at most 2. As a corollary, if $g(F)=2$, then there is at most one separating slope $\gamma$ on $F$ such that $M(\gamma )$ is either reducible or $\partial$-reducible.


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Additional Information

Ruifeng Qiu
Affiliation: Department of Applied Mathematics, Dalian University of Technology, Dalian, People’s Republic of China, 116022
Email: qiurf@dlut.edu.cn

Mingxing Zhang
Affiliation: Department of Applied Mathematics, Dalian University of Technology, Dalian, People’s Republic of China, 116022
Email: zhangmx@dlut.edu.cn

Keywords: Handle addition, Scharlemann cycle, virtual Scharlemann cycle
Received by editor(s): March 4, 2007
Published electronically: November 24, 2008
Additional Notes: This research was supported by NSFC(10625102) and a grant of SRFDP
Article copyright: © Copyright 2008 American Mathematical Society