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Connected sums of unstabilized Heegaard splittings are unstabilized. (English) Zbl 1152.57020
Let $$M_1$$ and $$M_2$$ be closed connected orientable $$3$$-manifolds, and let $$H_i$$ be a Heegaard surface of $$M_i$$. Then the connected sum $$H=H_1\sharp H_2$$ gives a Heegaard surface in $$M_1\sharp M_2$$. If one of the $$H_i$$ is stabilized, that is, the connected sum of some lower genus Heegaard surface and the standard genus one Heegaard surface of the $$3$$-sphere, then $$H$$ is also stabilized. C. McA. Gordon, in Kirby’s problem list [Kazez, William H. (ed.), Geometric topology. 1993 Georgia international topology conference, August 2–13, 1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2(pt.2), 35–473 (1997; Zbl 0888.57014)], conjectured that the converse is also true. The article under review gives a proof of this conjecture. (Independently, the same result is announced by Ruifeng Qiu.) In addition, it is shown that the prime decomposition of unstabilized Heegaard splittings is unique.
The proofs need a long and hard argument involving a sequence of generalized Heegaard splittings and a weak reduction process.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
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