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A sufficient condition for the genus of an amalgamated 3-manifold not to go down

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Abstract

Let M i be a connected, compact, orientable 3-manifold, F i a boundary component of M i with g(F i ) ⩾ 2, i = 1, 2, and F 1 ≊ F 2. Let φ: F 1 → F 2 be a homeomorphism, and M = M 1 ∪φ M 2, F = F 2 = φ(F 1). Then it is known that g(M) ⩽ g(M 1)+g(M 2)−g(F). In the present paper, we give a sufficient condition for the genus of an amalgamated 3-manifold not to go down as follows: Suppose that there is no essential surface with boundary (Q i , ∂Q i ) in (M i , F i ) satisfying χ(Q i ) > 3 − 2g(M i ), i = 1, 2. Then g(M) = g(M 1) + g(M 2) − g(F).

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References

  1. Bachman D, Derby-Talbot R. Degeneration of Heegaard genus, a survey. Geom Topol Monogr, 2007, 12: 1–15

    Article  MathSciNet  Google Scholar 

  2. Casson A J, Gordon C McA. Reducing Heegaard splittings. Topology Appl, 1987, 27: 275–283

    Article  MATH  MathSciNet  Google Scholar 

  3. Hempel J. 3-Manifolds. Princeton: Princeton University Press, 1976

    MATH  Google Scholar 

  4. Kobayashi T, Qiu R F. The amalgamation of high distance Heegaard splittings is always efficient. Math Ann, 2008, 341: 707–715

    Article  MATH  MathSciNet  Google Scholar 

  5. Kobayashi T, Qiu R F, Rieck Y, et al. Separating incompressible surfaces and stabilizations of Heegaard splittings. Math Proc Cambridge Philos Soc, 2004, 137: 633–643

    Article  MATH  MathSciNet  Google Scholar 

  6. Lackenby M. The Heegaard genus of amalgamated 3-manifolds. Geom Dedicata, 2004, 109: 139–145

    Article  MATH  MathSciNet  Google Scholar 

  7. Lei F C, Yang G Q. A lower bound of genus of amalgamtions of Heegaard splittings. Math Proc Cambridge Philos Soc, 2009, 146: 615–623

    Article  MATH  MathSciNet  Google Scholar 

  8. Li T. On the Heegaard splittings of amalgamated 3-manifolds. Geom Topol Monogr, 2007, 12: 157–190

    Article  Google Scholar 

  9. Scharlemann M. Local detection of strongly irreducible Heegaard splittings. Topology Appl, 1998, 90: 135–147

    Article  MATH  MathSciNet  Google Scholar 

  10. Scharlemann M, Thompson A. Thin position for 3-manifolds. Contemp Math, 1994, 164: 231–238

    MathSciNet  Google Scholar 

  11. Schultens J. The classification of Heegaard splittings for (compact orientable surfaces)×S 1. Proc London Math Soc, 1993, 67: 425–448

    Article  MATH  MathSciNet  Google Scholar 

  12. Schultens J. Additivity of tunnel number for small knots. Comment Math Helv, 2000, 75: 353–363

    Article  MATH  MathSciNet  Google Scholar 

  13. Schultens J, Weidmann R. Destabilizing amalgamated Heegaard splittings. Geom Topol Monogr, 2007, 12: 319–334

    Article  MathSciNet  Google Scholar 

  14. Souto J. Distance in the curve complex and Heegaard genus. Preprint

  15. Yang G Q, Lei F C. On amalgamtions of Heegaard splittings with high distance. Proc Amer Math Soc, 2009, 137: 723–731

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    FengLing Li & GuoQiu Yang

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    FengChun Lei

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  1. FengLing Li
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  2. GuoQiu Yang
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  3. FengChun Lei
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Correspondence to FengLing Li.

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Li, F., Yang, G. & Lei, F. A sufficient condition for the genus of an amalgamated 3-manifold not to go down. Sci. China Math. 53, 1697–1702 (2010). https://doi.org/10.1007/s11425-010-3130-8

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  • DOI: https://doi.org/10.1007/s11425-010-3130-8

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