zbMATH — the first resource for mathematics

A note on proof of Gordon’s conjecture. (English) Zbl 1397.57038
Let $$M$$ be a 3-manifold. If there is a closed surface $$S$$ which cuts $$M$$ into two compression bodies $$V$$ and $$W$$ with $$S =\partial_+ W = \partial_+V$$, then we say $$M$$ has a Heegaard splitting, denoted by $$M = V\cup_S W$$; and $$S$$ is called a Heegaard surface of $$M$$. If there is an essential disk in each of the two compression bodies, such that the two disks intersect in a single point, then the Heegaard splitting is said to be stabilized and we may find another Heegaard splitting of the 3-manifold with a lower genus. Now suppose $$M$$ is a reducible 3-manifold such that $$M$$=$$M_ 1\#M_2$$. There is a standard Heegaard splitting of $$M$$= $$V\cup_S W$$, called the connected sum of $$M_1 = V_1\cup_{S_1}W_1$$ and $$M_2 = V_2\cup_{S_2} W_2$$. Gordon conjectured that $$V\cup_S W$$ is stabilized if and only if one of $$M_1 = V_1\cup_{S_1}W_1$$ and $$M_2=V_2\cup_{S_2} W_2$$ are stabilized, and this conjecture has been proven by D. Bachman [Geom. Topol. 12, No. 4, 2327–2378 (2008; Zbl 1152.57020)] and R. Qiu and M. Scharlemann [Adv. Math. 222, No. 6, 2085–2106 (2009; Zbl 1180.57025)]. This paper gives an alternative proof of Gordon’s Conjecture by using Qiu’s labels and two new labels.
Reviewer: Qiang E (Dalian)

MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
Full Text:
References:
 [1] D. Bachman, Connected sums of unstabilized Heegaard splittings are unstabilized, Geom. Topol. 12 (2008), no. 4, 2327-2378. · Zbl 1152.57020 [2] R. Kirby, Problems in low dimensional manifold theory, in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, 273-312, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, RI, 1978. [3] T. Kobayashi and R. Qiu, The amalgamation of high distance Heegaard splittings is always efficient, Math. Ann. 341 (2008), no. 3, 707-715. · Zbl 1140.57012 [4] T. Kobayashi, R. Qiu, Y. Rieck, and S. Wang, Separating incompressible surfaces and stabilizations of Heegaard splittings, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 3, 633-643. · Zbl 1062.57028 [5] M. Lackenby, The Heegaard genus of amalgamated 3-manifolds, Geom. Dedicata 109 (2004), 139-145. · Zbl 1081.57018 [6] R. Qiu, Stabilizations of reducible Heegaard splittings, arXiv: math. GT/0409497. [7] R. Qiu and M. Scharlemann, A proof of the Gordon conjecture, Adv. Math. 222 (2009), no. 6, 2085-2106. · Zbl 1180.57025 [8] J. Schultens, The classification of Heegaard splittings for (compact orientable surface) ×S1, Proc. London Math. Soc. (3) 67 (1993), no. 2, 425-448. · Zbl 0789.57012 [9] J. Schultens and R. Weidmann, Destabilizing amalgamated Heegaard splittings, in Workshop on Heegaard Splittings, 319-334, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007. Kun Du School of Mathematics and Statistics Lanzhou University Lanzhou 730000, P. R. China Email address: dukun@lzu.edu.cn · Zbl 1216.57011
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.