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Unstabilized and uncritical self-amalgamation along essential subsurfaces

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Abstract

Suppose V ∪SW is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F1 and F2 are pairwise disjoint homeomorphic essential subsurfaces in ∂_V. In this paper, we give a suffcient condition such that the self-amalgamation of V ∪SW along F1 and F2 is unstabilized and uncritical.

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References

  1. Bachman D. Connected sums of unstabilized Heegaard splittings are unstabilized. Geom Topol, 2008, 12: 2327–2378

    Article  MathSciNet  MATH  Google Scholar 

  2. Bachman D. Topological index theory for surfaces in 3-manifolds. Geom Topol, 2010, 14: 585–609

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Bachman D, Schleimer S, Sedgwick E. Sweepouts of amalgamated 3-manifolds. Algebr Geom Topol, 2006, 6: 171–194

    Article  MathSciNet  MATH  Google Scholar 

  5. Du K, Gao X T. A note on Heegaard splittings of amalgamated 3-manifolds. Chin Ann Math Ser B, 2011, 32: 475–482

    Article  MathSciNet  MATH  Google Scholar 

  6. Du K, Qiu R F. The self-amalgamation of high distance Heegaard splittings is always effcient. Topology Appl, 2010, 157: 1136–1141

    Article  MathSciNet  MATH  Google Scholar 

  7. E Q, Lei F C. Critical Heegaard surfaces obtained by self-amalgamation. J Knot Theory Ramifications, 2013, 22: 1350015

    Article  MathSciNet  MATH  Google Scholar 

  8. Harvey W J. Boundary structure of the modular group. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Princeton: Princeton University Press, 1981, 245–251

    Chapter  Google Scholar 

  9. Hempel J. 3-Manifolds as viewed from curve complex. Topology, 2001, 40: 631–657

    Article  MathSciNet  MATH  Google Scholar 

  10. Kim J. On critical Heegaard splittings of tunnel number two composite knot exteriors. J Knot Theory Ramifications, 2013, 22: 1350065

    Article  MathSciNet  MATH  Google Scholar 

  11. Kim J. On unstabilized genus three critical Heegaard surfaces. Topology Appl, 2014, 165: 98–109

    Article  MathSciNet  MATH  Google Scholar 

  12. Kirby R. Problems in low-dimensional topology. Algebr Geom Topol, 1978, 32: 273–312

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee J. Critical Heegaard surfaces obtained by amalgamation. Topology Appl, 2013, 160: 111–116

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Li T. Heegaard surfaces and the distance of amalgamation. Geom Topol, 2010, 14: 1871–1919

    Article  MathSciNet  MATH  Google Scholar 

  19. Li T. Images of the disk complex. Geom Dedicata, 2012, 158: 121–136

    Article  MathSciNet  MATH  Google Scholar 

  20. Liang L, Lei F C, Li F L. A suffcient condition for a self-amalgamation of a Heegaard splitting to be unstabilized. Topology Appl, 2014, 178: 345–351

    Article  MathSciNet  MATH  Google Scholar 

  21. Masur H, Minsky Y. Geometry of the complex of curves, II: Hierarchical structure. Geom Funct Anal, 2000, 10: 902–974

    Article  MathSciNet  MATH  Google Scholar 

  22. Masur H, Schleimer S. The geometry of the disk complex. J Amer Math Soc, 2013, 26: 1–62

    Article  MathSciNet  MATH  Google Scholar 

  23. Qiu R F, Scharlemann M. A proof of the Gordon conjecture. Adv Math, 2009, 222: 2085–2106

    Article  MathSciNet  MATH  Google Scholar 

  24. Schultens J. The classification of Heegaard splittings for (compact orientable surface)×S1. Proc Lond Math Soc (3), 1993, 67: 425–448

    Article  MathSciNet  MATH  Google Scholar 

  25. Schultens J, Weidmann R. Destabilizing amalgamated Heegaard splittings. Geom Topol Monogr, 2007, 12: 319–344

    Article  MathSciNet  MATH  Google Scholar 

  26. Souto J. Distance in the curve complex and Heegaard genus. Https://sites.google.com/site/jsoutoc/papers

  27. Yang G Q, Lei F C. Amalgamations of Heegaard splittings in 3-manifolds without essential surfaces. Algebr Geom Topol, 2009, 9: 2041–2054

    Article  MathSciNet  MATH  Google Scholar 

  28. Zou Y Q, Du K, Guo Q L, et al. Unstabilized self-amalgamation of a Heegaard splitting. Topology Appl, 2013, 160: 406–411

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11601209, 11671064, 11471151, 11401069, 11329101 and 11431009), Scientific Research Foundation for Doctors of Liaoning Province (Grant No. 201601239), Scientific Research Fund of Liaoning Provincial Education Depart- ment (Grant No. L201683660) and the Fundamental Research Funds for the Central Universities (Grant No. DUT16LK40). The authors thank the referees for numerous suggestions that have improved the exposition of this paper.

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Authors and Affiliations

  1. School of Mathematics, Liaoning Normal University, Dalian, 116029, China

    Liang Liang

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

    Fengling Li & Fengchun Lei

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  1. Liang Liang
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  2. Fengling Li
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  3. Fengchun Lei
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Correspondence to Liang Liang.

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Liang, L., Li, F. & Lei, F. Unstabilized and uncritical self-amalgamation along essential subsurfaces. Sci. China Math. 62, 1807–1812 (2019). https://doi.org/10.1007/s11425-017-9202-3

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  • DOI: https://doi.org/10.1007/s11425-017-9202-3

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