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A refinement of Johnson’s bounding for the stable genera of Heegaard splittings. (English) Zbl 1220.57012

The stabilization conjecture is a well known problem within the topology of 3-manifolds (see Problem 3.89 in R. Kirby’s list [Problems in low-dimensional topology, AMS/IP Stud. Adv. Math. 2 (pt. 2), 35–473 (1997; Zbl 0888.57014)]): it states that the minimal genus of a common stabilization of two Heegaard splittings of the same 3-manifold \(M\) (i.e. the so called stable genus) is at most \(p+1\), \(p\) being the larger of the two initial genera.
The conjecture has been verified for many classes of 3-manifolds: see [J. Schultens, Topology Appl. 73, No. 2, 133–139 (1996; Zbl 0867.57013)], [H. Rubinstein and M. Scharleman, Geom. Topol. Monogr. 2, 489–553 (1999; Zbl 0962.57013)], [R. Derby-Talbot, Topology Appl. 154, No. 9, 1841–1853 (2007; Zbl 1143.57009)].
In [Geom. Topol. 13, No. 4, 2029–2050 (2009; Zbl 1177.57018)], J. Hass, A. Thompson and W. Thurston gave a counterexample for the “oriented version” of the conjecture. In [D. Bachman, “Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization”, arXiv:math/0903.1695] both similar results with different techniques and a counterexample for the general conjecture were obtained. Finally, in [J. Topol. 3, No. 3, 668–690 (2010; Zbl 1246.57044)], J. Johnson gave a class of counterexamples by constructing, for each \(k \geq 2\), an irreducible toroidal 3-manifold with Heegaard splittings of genera \(2k+1\) and \(2k\) such that the stable genus of these splittings is \(3k+1.\)
The present paper refines the bounding of the stable genus from below: the main theorem states that, for every \(k \geq 2\), there exists a manifold with two Heegaard splittings of genus \(2k\) whose stable genus is \(3k\).
The proof is obtained by modifying Johnson’s construction: in fact, while Johnson makes use of amalgamations of two Heegaard splittings with high distance along the torus boundaries, the author takes into account sphere boundaries. The result follows by proving that amalgamation along sphere boundaries is nothing else than connected sum, and from Hempel’s construction of Heegaard splittings with arbitrarily high distance [see J. Hempel, Topology 40, No. 3, 631–657 (2001; Zbl 0985.57014)].

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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References:

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