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)].


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


[1] D. Bachman: Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization , · Zbl 1260.57036
[2] J. Berge and M. Scharlemann: Multiple genus 2 Heegaard splittings: a missed case , · Zbl 1232.57020
[3] J. Cerf: Sur les Difféomorphismes de la Sphère de Dimension Trois \((\Gamma_{4}=0)\), Lecture Notes in Mathematics 53 Springer, Berlin, 1968. · Zbl 0164.24502 · doi:10.1007/BFb0060395
[4] R. Derby-Talbot: Stabilizing Heegaard splittings of toroidal 3-manifolds , Topology Appl. 154 (2007), 1841-1853. · Zbl 1143.57009 · doi:10.1016/j.topol.2007.01.011
[5] J. Hass, A. Thompson and W. Thurston: Stabilization of Heegaard splittings , Geom. Topol. 13 (2009), 2029-2050. · Zbl 1177.57018 · doi:10.2140/gt.2009.13.2029
[6] M.W. Hirsch: Differential Topology, Graduate Texts in Mathematics 33 , Springer, New York, 1994.
[7] J. Hempel: 3-manifolds as viewed from the curve complex ; Topology 40 (2001), 631-657. · Zbl 0985.57014 · doi:10.1016/S0040-9383(00)00033-1
[8] J. Johnson: Flipping and stabilizing Heegaard splittings ,
[9] J. Johnson: Bounding the stable genera of Heegaard splittings from below , · Zbl 1246.57044 · doi:10.1112/jtopol/jtq021
[10] T. Kobayashi: Structures of the Haken manifolds with Heegaard splittings of genus two , Osaka J. Math. 21 (1984), 437-455. · Zbl 0568.57007
[11] T. Kobayashi and O. Saeki: The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map , Pacific J. Math. 195 (2000), 101-156. · Zbl 1019.57010 · doi:10.2140/pjm.2000.195.101
[12] K. Reidemeister: Zur dreidimensionalen Topologie , Abh. Math. Sem. Univ. Hamburg 11 (1933), 189-194. · Zbl 0007.08005 · doi:10.1007/BF02940644
[13] H. Rubinstein and M. Scharlemann: Comparing Heegaard splittings of non-Haken \(3\)-manifolds , Topology 35 (1996), 1005-1026. · Zbl 0858.57020 · doi:10.1016/0040-9383(95)00055-0
[14] H. Rubinstein and M. Scharlemann: Genus two Heegaard splittings of orientable three-manifolds ; in Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr. 2 , Geom. Topol. Publ., Coventry, 1999, 489-553. · Zbl 0962.57013
[15] J. Schultens: The stabilization problem for Heegaard splittings of Seifert fibered spaces , Topology Appl. 73 (1996), 133-139. · Zbl 0867.57013 · doi:10.1016/0166-8641(96)00030-2
[16] E. Sedgwick: An infinite collection of Heegaard splittings that are equivalent after one stabilization , Math. Ann. 308 (1997), 65-72. · Zbl 0873.57010 · doi:10.1007/s002080050064
[17] J. Singer: Three-dimensional manifolds and their Heegaard diagrams , Trans. Amer. Math. Soc. 35 (1933), 88-111. JSTOR: · Zbl 0006.18501 · doi:10.2307/1989314
[18] F. Waldhausen: On irreducible \(3\)-manifolds which are sufficiently large , Ann. of Math. (2) 87 (1968), 56-88. JSTOR: · Zbl 0157.30603 · doi:10.2307/1970594
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