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Transactions of the American Mathematical Society

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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.


References [Enhancements On Off] (What's this?)

  • 1. M. Culler, C. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. Math. (2), 125 (1987), 237-300. MR 881270 (88a:57026)
  • 2. C. Gordon, Small surfaces and Dehn fillings,
    Geom. Topol. Monogr., 2 (1999), 177-199. MR 1734408 (2000j:57036)
  • 3. C. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology, 35 (1996), 385-409. MR 1380506 (97b:57013)
  • 4. A. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math., 99 (1982), 373-377. MR 658066 (83h:57016)
  • 5. C. Hayashi and K. Motegi, Only single twists on unknots can produce composite knots, Trans. Amer. Math. Soc., 349 (1997), 4465-4479. MR 1355073 (98b:57010b)
  • 6. Y. Li, R. Qiu and M. Zhang, Boundary reducible handle additions on simple $ 3$-manifolds, to appear in Acta Math. Sin. (Engl. Ser.).
  • 7. R. Qiu, Reducible Dehn surgery and annular Dehn surgery, Pacific J. Math., 192 (2000), 357-368. MR 1744575 (2001b:57036)
  • 8. R. Qiu and S. Wang, Small knots and large handle additions, Comm. Anal. Geom., 13 (2005), 939-961. MR 2216147 (2007d:57020)
  • 9. R. Qiu and S. Wang, Handle additions producing essential closed surfaces, Pacific J. Math., 229 (2007), 233-255. MR 2276510 (2008e:57007)
  • 10. M. Scharlemann and Y. Wu, Hyperbolic manifolds and degenerating handle additions, J. Austral. Math. Soc. Ser. A 55 (1993), 72-89. MR 1231695 (94e:57019)
  • 11. M. Zhang, R. Qiu and Y. Li, The distance between two separating, reducing slopes is at most $ 4$, Math. Z., 257 (2007), 799-810. MR 2342554
  • 12. Y. Wu, The reducibility of surgered $ 3$-manifolds, Topology Appl., 43 (1992), 213-218. MR 1158868 (93e:57032)

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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: https://doi.org/10.1090/S0002-9947-08-04761-2
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

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