On amalgamations of Heegaard splittings with high distance

Yang, Guoqiu; Lei, Fengchun

doi:10.1090/S0002-9939-08-09642-1

On amalgamations of Heegaard splittings with high distance
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by Guoqiu Yang and Fengchun Lei PDF

Proc. Amer. Math. Soc. 137 (2009), 723-731 Request permission

Abstract:

Let $M$ be a compact, orientable 3-manifold and $F$ an essential closed surface which cuts $M$ into $M_{1}$ and $M_{2}$. Suppose that $M_{i}$ has a Heegaard splitting $V_{i}\cup _{S_{i}}W_{i}$ with distance $D{(S_{i})}\geqslant {2g(M_{i})+1}$, $i=1, 2$. Then $g(M)=g(M_1)+g(M_2)-g(F)$, and the amalgamation of $V_{1}\cup _{S_{1}}W_{1}$ and $V_{2}\cup _{S_{2}}W_{2}$ is the unique minimal Heegaard splitting of $M$ up to isotopy.

References

D. Bachman and R. Derby-Talbot, Degeneration of Heegaard genus, a survey, arXiv:math.GT/0606383v3, preprint.
David Bachman, Saul Schleimer, and Eric Sedgwick, Sweepouts of amalgamated 3-manifolds, Algebr. Geom. Topol. 6 (2006), 171–194. MR 2199458, DOI 10.2140/agt.2006.6.171
A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275–283. MR 918537, DOI 10.1016/0166-8641(87)90092-7
Kevin Hartshorn, Heegaard splittings of Haken manifolds have bounded distance, Pacific J. Math. 204 (2002), no. 1, 61–75. MR 1905192, DOI 10.2140/pjm.2002.204.61
John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
John Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631–657. MR 1838999, DOI 10.1016/S0040-9383(00)00033-1
William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
Tsuyoshi Kobayashi, Ruifeng Qiu, Yo’av Rieck, and Shicheng Wang, Separating incompressible surfaces and stabilizations of Heegaard splittings, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 3, 633–643. MR 2103921, DOI 10.1017/S0305004104007790
T. Kobayashi and R. Qiu, The amalgamation of high distance Heegaard splittings is always efficient, Math. Ann., Online: DOI 10.1007/s00208-008-0214-7.
Marc Lackenby, The Heegaard genus of amalgamated 3-manifolds, Geom. Dedicata 109 (2004), 139–145. MR 2113191, DOI 10.1007/s10711-004-6553-y
T. Li, On the Heegaard splittings of amalgamated $3$-manifolds, arXiv:math.GT/0701395, preprint.
Martin Scharlemann and Abigail Thompson, Thin position for $3$-manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231–238. MR 1282766, DOI 10.1090/conm/164/01596
Martin Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998), no. 1-3, 135–147. MR 1648310, DOI 10.1016/S0166-8641(97)00184-3
Martin Scharlemann and Maggy Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006), 593–617. MR 2224466, DOI 10.2140/gt.2006.10.593
Jennifer Schultens, Additivity of tunnel number for small knots, Comment. Math. Helv. 75 (2000), no. 3, 353–367. MR 1793793, DOI 10.1007/s000140050131
Jennifer Schultens, The classification of Heegaard splittings for (compact orientable surface)$\,\times \, S^1$, Proc. London Math. Soc. (3) 67 (1993), no. 2, 425–448. MR 1226608, DOI 10.1112/plms/s3-67.2.425
J. Souto, Distance in the curve complex and Heegaard genus, preprint.