DOI QR코드

DOI QR Code

A LOWER BOUND FOR THE GENUS OF SELF-AMALGAMATION OF HEEGAARD SPLITTINGS

  • Li, Fengling (DEPARTMENT OF MATHEMATICS HARBIN INSTITUTE OF TECHNOLOGY) ;
  • Lei, Fengchun (SCHOOL OF MATHEMATICS DALIAN UNIVERSITY OF TECHNOLOGY)
  • Received : 2009.05.11
  • Published : 2011.01.31

Abstract

Let M be a compact orientable closed 3-manifold, and F a non-separating incompressible closed surface in M. Let M' = M - ${\eta}(F)$, where ${\eta}(F)$ is an open regular neighborhood of F in M. In the paper, we give a lower bound of genus of self-amalgamation of minimal Heegaard splitting $V'\;{\cup}_{S'}\;W'$ of M' under some conditions on the distance of the Heegaard splitting.

Keywords

References

  1. D. Bachman and R. Derby-Talbot, Degeneration of Heegaard genus, a survey, Workshop on Heegaard Splittings, 1-15, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.
  2. D. Bachman, S. Schleimer, and E. Sedgwick, Sweepouts of amalgamated 3-manifolds, Algebr. Geom. Topol. 6 (2006), 171-194. https://doi.org/10.2140/agt.2006.6.171
  3. A. J. Casson and C. McA Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275-283. https://doi.org/10.1016/0166-8641(87)90092-7
  4. K. Du, F. Lei, and J. Ma, Distance and self-amalgamation of Heegaard splittings, preprint.
  5. K. Hartshorn, Heegaard splittings of Haken manifolds have bounded distance, Pacific J. Math. 204 (2002), no. 1, 61-75. https://doi.org/10.2140/pjm.2002.204.61
  6. J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631-657. https://doi.org/10.1016/S0040-9383(00)00033-1
  7. T. Kobayashi and R. Qiu, The amalgamation of high distance Heegaard splittings is always efficient, Math. Ann. 341 (2008), no. 3, 707-715. https://doi.org/10.1007/s00208-008-0214-7
  8. T. Kobayashi, R. Qiu, Y. Rieck, and S. Wang, Separating incompressible surfaces and stabilizations of Heegaard splittings, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 3, 633-643. https://doi.org/10.1017/S0305004104007790
  9. M. Lackenby, The Heegaard genus of amalgamated 3-manifolds, Geom. Dedicata 109 (2004), 139-145. https://doi.org/10.1007/s10711-004-6553-y
  10. T. Li, On the Heegaard splittings of amalgamated 3-manifolds, Workshop on Heegaard Splittings, 157-190, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.
  11. Y. Moriah, On boundary primitive manifolds and a theorem of Casson-Gordon, Topology Appl. 125 (2002), no. 3, 571-579. https://doi.org/10.1016/S0166-8641(01)00303-0
  12. K. Morimoto, Tunnel number, connected sum and meridional essential surfaces, Topology 39 (2000), no. 3, 469-485. https://doi.org/10.1016/S0040-9383(98)00070-6
  13. R. Qiu and F. Lei, On the Heegaard genera of 3-manifolds containing non-separating surfaces, Topology and physics, 341-347, Nankai Tracts Math., 12, World Sci. Publ., Hackensack, NJ, 2008.
  14. M. Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998), no. 1-3, 135-147. https://doi.org/10.1016/S0166-8641(97)00184-3
  15. M. Scharlemann and A. Thompson, Thin position for 3-manifolds, Geometric topology (Haifa, 1992), 231-238, Contemp. Math., 164, Amer. Math. Soc., Providence, RI, 1994.
  16. M. Scharlemann and A. Thompson, Heegaard splittings of (surface) ${\times}$ I are standard, Math. Ann. 295 (1993), no. 3, 549-564. https://doi.org/10.1007/BF01444902
  17. M. Scharlemann and M. Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006), 593-617. https://doi.org/10.2140/gt.2006.10.593
  18. J. Schultens, Additivity of tunnel number for small knots, Comment. Math. Helv. 75 (2000), no. 3, 353-367. https://doi.org/10.1007/s000140050131
  19. J. Schultens and R. Weidmann, Destabilizing amalgamated Heegaard splittings, Workshop on Heegaard Splittings, 319-334, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.
  20. J. Souto, Distance in the curve complex and Heegaard genus, preprint.
  21. G. Yang and F. Lei, On amalgamations of Heegaard splittings with high distance, Proc. Amer. Math. Soc. 137 (2009), no. 2, 723-731. https://doi.org/10.1090/S0002-9939-08-09642-1