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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)
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##### References:
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