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Transactions of the American Mathematical Society
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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04761-2
PII: S 0002-9947(08)04761-2
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