## Distance degenerating handle additions

HTML articles powered by AMS MathViewer

- by Liang Liang, Fengchun Lei and Fengling Li PDF
- Proc. Amer. Math. Soc.
**144**(2016), 423-434 Request permission

## Abstract:

Let $M=V\cup _{S}W$ be a Heegaard splitting of a 3-manifold $M$ and let $F$ be a component of $\partial M$ lying in $\partial _{-}V$. A simple closed curve $J$ in $F$ is said to be distance degenerating if the distance of $M_{J}=V_{J}\cup _{S}W$ is less than the distance of $M=V\cup _{S}W$ where $M_{J}$ is the 3-manifold obtained by attaching a 2-handle to $M$ along $J$. In this paper, we will prove that for a strongly irreducible Heegaard splitting $M=V\cup _{S}W$, if $V$ is simple or $M=V\cup _{S}W$ is locally complicated, then the diameter of the set of distance degenerating curves in $F$ is bounded.## References

- Ayako Ido, Yeonhee Jang, and Tsuyoshi Kobayashi,
*Heegaard splittings of distance exactly $n$*, Algebr. Geom. Topol.**14**(2014), no. 3, 1395–1411. MR**3190598**, DOI 10.2140/agt.2014.14.1395 - 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 - 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 - 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 - W. J. Harvey,
*Boundary structure of the modular group*, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 245–251. MR**624817** - William Jaco,
*Adding a $2$-handle to a $3$-manifold: an application to property $R$*, Proc. Amer. Math. Soc.**92**(1984), no. 2, 288–292. MR**754723**, DOI 10.1090/S0002-9939-1984-0754723-6 - Tsuyoshi Kobayashi and Ruifeng Qiu,
*The amalgamation of high distance Heegaard splittings is always efficient*, Math. Ann.**341**(2008), no. 3, 707–715. MR**2399167**, DOI 10.1007/s00208-008-0214-7 - Feng Chun Lei,
*A proof of Przytycki’s conjecture on $n$-relator $3$-manifolds*, Topology**34**(1995), no. 2, 473–476. MR**1318887**, DOI 10.1016/0040-9383(95)93238-3 - Fengchun Lei,
*A general handle addition theorem*, Math. Z.**221**(1996), no. 2, 211–216. MR**1376293**, DOI 10.1007/PL00004514 - Tao Li,
*Images of the disk complex*, Geom. Dedicata**158**(2012), 121–136. MR**2922707**, DOI 10.1007/s10711-011-9624-x - H. A. Masur and Y. N. Minsky,
*Geometry of the complex of curves. II. Hierarchical structure*, Geom. Funct. Anal.**10**(2000), no. 4, 902–974. MR**1791145**, DOI 10.1007/PL00001643 - Howard Masur and Saul Schleimer,
*The geometry of the disk complex*, J. Amer. Math. Soc.**26**(2013), no. 1, 1–62. MR**2983005**, DOI 10.1090/S0894-0347-2012-00742-5 - Martin Lustig and Yoav Moriah,
*Horizontal Dehn surgery and genericity in the curve complex*, J. Topol.**3**(2010), no. 3, 691–712. MR**2684517**, DOI 10.1112/jtopol/jtq022 - Józef H. Przytycki,
*Incompressibility of surfaces after Dehn surgery*, Michigan Math. J.**30**(1983), no. 3, 289–308. MR**725782**, DOI 10.1307/mmj/1029002906 - Józef H. Przytycki,
*$n$-relator $3$-manifolds with incompressible boundary*, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 273–285. MR**903870** - Martin Scharlemann,
*Proximity in the curve complex: boundary reduction and bicompressible surfaces*, Pacific J. Math.**228**(2006), no. 2, 325–348. MR**2274524**, DOI 10.2140/pjm.2006.228.325 - 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 - Martin Scharlemann and Ying Qing Wu,
*Hyperbolic manifolds and degenerating handle additions*, J. Austral. Math. Soc. Ser. A**55**(1993), no. 1, 72–89. MR**1231695** - Guoqiu Yang and Fengchun Lei,
*On amalgamations of Heegaard splittings with high distance*, Proc. Amer. Math. Soc.**137**(2009), no. 2, 723–731. MR**2448595**, DOI 10.1090/S0002-9939-08-09642-1 - Fengchun Lei and Guoqiu Yang,
*A lower bound of genus of amalgamations of Heegaard splittings*, Math. Proc. Cambridge Philos. Soc.**146**(2009), no. 3, 615–623. MR**2496347**, DOI 10.1017/S030500410800203X - Yanqing Zou, Kun Du, Qilong Guo, and Ruifeng Qiu,
*Unstabilized self-amalgamation of a Heegaard splitting*, Topology Appl.**160**(2013), no. 2, 406–411. MR**3003339**, DOI 10.1016/j.topol.2012.11.020

## Additional Information

**Liang Liang**- Affiliation: School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China
- Address at time of publication: School of Mathematics, Liaoning Normal University, Dalian 116029, People’s Republic of China
- Email: liang_liang@aliyun.com
**Fengchun Lei**- Affiliation: School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China
- Email: fclei@dlut.edu.cn
**Fengling Li**- Affiliation: School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China
- MR Author ID: 893090
- Email: dutlfl@163.com
- Received by editor(s): June 26, 2014
- Received by editor(s) in revised form: September 2, 2014, December 7, 2014, and December 8, 2014
- Published electronically: June 24, 2015
- Additional Notes: The second author was supported by the Fundamental Research Funds for the Central Universities (No. DUT14ZD208) and partially supported by grant No.11329101 of NSFC

The third author was supported by the Fundamental Research Funds for the Central Universities (No. DUT14LK12) and partially supported by two grants No.11101058 and No.11329101 of NSFC - Communicated by: Martin Scharlemann
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 423-434 - MSC (2010): Primary 57N10; Secondary 57M50
- DOI: https://doi.org/10.1090/proc/12688
- MathSciNet review: 3415608