# zbMATH — the first resource for mathematics

The amalgamation of high distance Heegaard splittings is always efficient. (English) Zbl 1140.57012
Summary: Let $$M$$ be a compact orientable 3-manifold, and $$F$$ be an essential closed surface which cuts $$M$$ into two 3-manifolds $$M_{1}$$ and $$M_{2}$$. Let $${M_{i}=V_{i}\cup_{S_{i}} W_{i}}$$ be a Heegaard splitting for $$i = 1, 2$$. We denote by $$d(S_{i})$$ the distance of $${V_{i}\cup_{S_{i}} W_{i}}$$. If $$d(S_{1}), d(S_{2}) \geq 2(g (M_{1}) + g (M_{2}) - g (F))$$, then $$M$$ has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of $${V_{1}\cup_{S_{1}} W_{1}}$$ and $${V_{2}\cup_{S_{2}} W_{2}}$$.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
Full Text:
##### References:
 [1] Bachman, D.: Connected sums of unstabilized Heegaard splittings are unstabilized, preprint, ArXiv: math.GT/0404058 · Zbl 1152.57020 [2] Bachman, D., Schleimer, S., Sedgwick, E.: Sweepouts of amalgamated 3-manifolds. Algebr. Geom. Topol. 6, 171–194 (2006) · Zbl 1099.57016 · doi:10.2140/agt.2006.6.171 [3] Casson, A., McA Gordon, C.: Reducing Heegaard splittings. Topol. Appl. 27, 275–283 (1987) · Zbl 0632.57010 · doi:10.1016/0166-8641(87)90092-7 [4] Hartshorn, K.: Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math. 204, 61–75 (2002) · Zbl 1065.57021 · doi:10.2140/pjm.2002.204.61 [5] Hempel, J.: 3-manifolds as viewed from the curve complex. Topology 40, 631–657 (2001) · Zbl 0985.57014 · doi:10.1016/S0040-9383(00)00033-1 [6] Kirby, R.: Problems in low-dimensional topology, Geometric topology, AMS/IP Stud. Adv. Math., vol. 2.2, pp. 35–473 (Athens, GA, 1993). Amer. Math. Soc., Providence, RI (1993) [7] Kobayashi, T., Qiu, R., Rieck, Y., Wang, S.: Separating incompressible surfaces and stabilizations of Heegaard splittings. Math. Proc. Cambridge Philos. Soc. 137, 633–643 (2004) · Zbl 1062.57028 · doi:10.1017/S0305004104007790 [8] Lackenby, M.: The Heegaard genus of amalgamated 3-manifolds. Geom. Dedicata 109, 139–145 (2004) · Zbl 1081.57018 · doi:10.1007/s10711-004-6553-y [9] Qiu, R.: Stabilizations of reducible Heegaard splittings, preprint, ArXiv: math.GT/0409497 [10] Scharlemann, M.: Proximity in the curve complex: boundary reduction and bicompressible surfaces. Pacific J. Math. 228, 325–348 (2006) · Zbl 1127.57010 · doi:10.2140/pjm.2006.228.325 [11] Scharlemann, M., Thompson, A.: Thin position for 3-manifolds. Geometric Topology (Haifa, 1992), Contemp. Math., vol. 164, pp. 231–238. Amer. Math. Soc., Providence, RI (1994) · Zbl 0818.57013 [12] Scharlemann, M., Thompson, A.: Heegaard splittings of Surfaces $$\times$$ I are standard. Math. Ann. 295, 549–564 (1993) · Zbl 0814.57010 · doi:10.1007/BF01444902 [13] Scharlemann, M., Tomova, M.: Alternate Heegaard genus bounds distance. Geom. Topol. 10, 593–617 (2006) · Zbl 1128.57022 · doi:10.2140/gt.2006.10.593 [14] Schultens, J.: Additivity of Tunnel number for small knots. Comment. Math. Helv. 75, 353–367 (2000) · Zbl 0972.57007 · doi:10.1007/s000140050131 [15] Schultens, J., Weidman R.: Destabilizing amalgamated Heegaard splittings, preprint, ArXiv: math. GT/0510386
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.