Distance degenerating handle additions

Liang, Liang; Lei, Fengchun; Li, Fengling

doi:10.1090/proc/12688

Distance degenerating handle additions
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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