On amalgamations of Heegaard splittings with high distance

Authors:
Guoqiu Yang and Fengchun Lei

Journal:
Proc. Amer. Math. Soc. **137** (2009), 723-731

MSC (2000):
Primary 57M99

Published electronically:
September 9, 2008

MathSciNet review:
2448595

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact, orientable 3-manifold and an essential closed surface which cuts into and . Suppose that has a Heegaard splitting with distance , . Then , and the amalgamation of and is the unique minimal Heegaard splitting of up to isotopy.

**1.**D. Bachman and R. Derby-Talbot,*Degeneration of Heegaard genus, a survey*, arXiv:math.GT/0606383v3, preprint.**2.**David Bachman, Saul Schleimer, and Eric Sedgwick,*Sweepouts of amalgamated 3-manifolds*, Algebr. Geom. Topol.**6**(2006), 171–194 (electronic). MR**2199458**, 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. MR**918537**, 10.1016/0166-8641(87)90092-7**4.**Kevin Hartshorn,*Heegaard splittings of Haken manifolds have bounded distance*, Pacific J. Math.**204**(2002), no. 1, 61–75. MR**1905192**, 10.2140/pjm.2002.204.61**5.**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****6.**John Hempel,*3-manifolds as viewed from the curve complex*, Topology**40**(2001), no. 3, 631–657. MR**1838999**, 10.1016/S0040-9383(00)00033-1**7.**William Jaco,*Lectures on three-manifold topology*, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR**565450****8.**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**, 10.1017/S0305004104007790**9.**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.**10.**Marc Lackenby,*The Heegaard genus of amalgamated 3-manifolds*, Geom. Dedicata**109**(2004), 139–145. MR**2113191**, 10.1007/s10711-004-6553-y**11.**T. Li,*On the Heegaard splittings of amalgamated -manifolds*, arXiv:math.GT/0701395, preprint.**12.**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**, 10.1090/conm/164/01596**13.**Martin Scharlemann,*Local detection of strongly irreducible Heegaard splittings*, Topology Appl.**90**(1998), no. 1-3, 135–147. MR**1648310**, 10.1016/S0166-8641(97)00184-3**14.**Martin Scharlemann and Maggy Tomova,*Alternate Heegaard genus bounds distance*, Geom. Topol.**10**(2006), 593–617 (electronic). MR**2224466**, 10.2140/gt.2006.10.593**15.**Jennifer Schultens,*Additivity of tunnel number for small knots*, Comment. Math. Helv.**75**(2000), no. 3, 353–367. MR**1793793**, 10.1007/s000140050131**16.**Jennifer Schultens,*The classification of Heegaard splittings for (compact orientable surface)×𝑆¹*, Proc. London Math. Soc. (3)**67**(1993), no. 2, 425–448. MR**1226608**, 10.1112/plms/s3-67.2.425**17.**J. Souto,*Distance in the curve complex and Heegaard genus*, preprint.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
57M99

Retrieve articles in all journals with MSC (2000): 57M99

Additional Information

**Guoqiu Yang**

Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang Province, People’s Republic of China

Email:
gqyang@hit.edu.cn

**Fengchun Lei**

Affiliation:
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, Liaoning Province, People’s Republic of China

Email:
ffcclei@yahoo.com.cn

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09642-1

Keywords:
Amalgamation,
distance of Heegaard splitting,
minimal Heegaard splitting

Received by editor(s):
August 6, 2007

Published electronically:
September 9, 2008

Additional Notes:
The second author is supported in part by a grant (No. 15071034) of NFSC and a grant (No. 893322) of DLUT

Communicated by:
Daniel Ruberman

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.