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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)

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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