an:06899177
Zbl 1397.57038
Du, Kun
A note on proof of Gordon's conjecture
EN
Bull. Korean Math. Soc. 55, No. 3, 699-715 (2018).
00406492
2018
j
57N10 57M50 57M27
3-manifolds; Heegaard splitting; stabilization; band sum; Gordon's conjecture
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 \textit{D. Bachman} [Geom. Topol. 12, No. 4, 2327--2378 (2008; Zbl 1152.57020)] and \textit{R. Qiu} and \textit{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.
Qiang E (Dalian)
Zbl 1152.57020; Zbl 1180.57025