×
Li, Tao

Rank and genus of 3-manifolds. (English) Zbl 1277.57004

J. Am. Math. Soc. 26, No. 3, 777-829 (2013).
In the 1960’s, F. Waldhausen conjectured the equality, for each closed orientable 3-manifold \(M\), between the rank \(r(M)\) of the fundamental group \(\pi_1(M)\) and the Heegaard genus \(g(M)\) of \(M\) [Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 2, 313–322 (1978; Zbl 0397.57007)]. Various counterexamples may be found in the literature: in particular, B. Boileau and H. Zieschang obtained a Seifert fibered space with \(r(M)=2\) and \(g(M)=3\) [Invent. Math. 76, 455–468 (1984; Zbl 0538.57004)], while J. Schultens and R. Weidman proved the existence of graph manifolds with discrepancy \(g(M) - r(M)\) arbitrarily large [Pac. J. Math. 231 (2), 481–510 (2007; Zbl 1171.57020)].
In 2007, the above Conjecture has been re-formulated, by restricting the attention to hyperbolic 3-manifolds: see [P. B. Shalen, in: Gordon, Cameron (ed.) et al., Proceedings of the Technion workshop on Heegaard splittings, Haifa, Israel, summer 2005. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 12, 335–349 (2007; Zbl 1140.57009)].
The present paper gives a negative answer to this “modern version” of the Waldhausen Conjecture. In fact, a closed orientable hyperbolic 3-manifold \(M\) is proved to exist, so that \(g(M) - r(M)\) is arbitrarily large.
Actually, the author produces an (atoroidal) 3-manifold with boundary \(\bar M\) with \(r(\bar M) < g(\bar M)\), and the closed counterexample is constructed starting from \(\bar M\), via the so called annulus sum see [the author, Geom. Topol. 14, No. 4, 1871–1919 (2010; Zbl 1207.57031)]. Moreover, every 2-bridge knot exterior is proved to be a JSJ piece of a closed 3-manifold \(M\) with \(r(M) < g(M).\)
Reviewer: Maria Rita Casali (Modena)

Cited in 23 Documents

MSC:

57M05 Fundamental group, presentations, free differential calculus
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)

Keywords:

Heegaard genus; rank; hyperbolic 3-manifold; annulus sum

Citations:

Zbl 0397.57007; Zbl 0538.57004; Zbl 1171.57020; Zbl 1140.57009; Zbl 1207.57031
PDF BibTeX XML Cite
\textit{T. Li}, J. Am. Math. Soc. 26, No. 3, 777--829 (2013; Zbl 1277.57004)
Full Text: DOI arXiv

References:

[1] Miklós Abért and Nikolay Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus problem. arXiv:math/0701361 · Zbl 1271.57046
[2] David Bachman, Saul Schleimer, and Eric Sedgwick, Sweepouts of amalgamated 3-manifolds, Algebr. Geom. Topol. 6 (2006), 171 – 194. · Zbl 1099.57016
[3] M. Boileau and H. Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds, Invent. Math. 76 (1984), no. 3, 455 – 468. · Zbl 0538.57004
[4] A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275 – 283. · Zbl 0632.57010
[5] I. A. Grushko, On the bases of a free product of groups. Matematicheskii Sbornik 8 (1940), 169-182.
[6] Wolfgang Haken, Some results on surfaces in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1968, pp. 39 – 98.
[7] Wolfgang Haken, Various aspects of the three-dimensional Poincaré problem, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 140 – 152.
[8] A. E. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), no. 2, 373 – 377. · Zbl 0502.57005
[9] Allen Hatcher, Basic Topology of Three Manifolds. Online notes, available at http://www.math.cornell.edu/hatcher/. · Zbl 1044.55001
[10] A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. 79 (1985), no. 2, 225 – 246. · Zbl 0602.57002
[11] John Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631 – 657. · Zbl 0985.57014
[12] Tsuyoshi Kobayashi, Classification of unknotting tunnels for two bridge knots, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 259 – 290. · Zbl 0962.57003
[13] Marc Lackenby, The Heegaard genus of amalgamated 3-manifolds, Geom. Dedicata 109 (2004), 139 – 145. · Zbl 1081.57018
[14] Marc Lackenby, The asymptotic behaviour of Heegaard genus, Math. Res. Lett. 11 (2004), no. 2-3, 139 – 149. · Zbl 1063.57002
[15] Tao Li, Heegaard surfaces and measured laminations. I. The Waldhausen conjecture, Invent. Math. 167 (2007), no. 1, 135 – 177. · Zbl 1109.57012
[16] Tao Li, Heegaard surfaces and measured laminations. II. Non-Haken 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 3, 625 – 657. · Zbl 1108.57015
[17] Tao Li, Saddle tangencies and the distance of Heegaard splittings, Algebr. Geom. Topol. 7 (2007), 1119 – 1134. · Zbl 1134.57005
[18] Tao Li, On the Heegaard splittings of amalgamated 3-manifolds, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 157 – 190. · Zbl 1153.57016
[19] Tao Li, Heegaard surfaces and the distance of amalgamation, Geom. Topol. 14 (2010), no. 4, 1871 – 1919. · Zbl 1207.57031
[20] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. · Zbl 0997.20037
[21] Yoav Moriah and Hyam Rubinstein, Heegaard structures of negatively curved 3-manifolds, Comm. Anal. Geom. 5 (1997), no. 3, 375 – 412. · Zbl 0890.57025
[22] Hossein Namazi and Juan Souto, Heegaard splittings and pseudo-Anosov maps, Geom. Funct. Anal. 19 (2009), no. 4, 1195 – 1228. · Zbl 1210.57019
[23] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 · Zbl 1130.53001
[24] Grisha Perelman, Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 · Zbl 1130.53002
[25] Grisha Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245 · Zbl 1130.53003
[26] Yo’av Rieck, Heegaard structures of manifolds in the Dehn filling space, Topology 39 (2000), no. 3, 619 – 641. · Zbl 0944.57013
[27] Yo’av Rieck and Eric Sedgwick, Persistence of Heegaard structures under Dehn filling, Topology Appl. 109 (2001), no. 1, 41 – 53. · Zbl 0968.57018
[28] Martin Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998), no. 1-3, 135 – 147. · Zbl 0926.57018
[29] Martin Scharlemann, Proximity in the curve complex: boundary reduction and bicompressible surfaces, Pacific J. Math. 228 (2006), no. 2, 325 – 348. · Zbl 1127.57010
[30] Martin Scharlemann, Heegaard splittings of compact 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 921 – 953. · Zbl 0985.57005
[31] Martin Scharlemann and Jennifer Schultens, The tunnel number of the sum of \? knots is at least \?, Topology 38 (1999), no. 2, 265 – 270. · Zbl 0929.57003
[32] Martin Scharlemann and Maggy Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006), 593 – 617. · Zbl 1128.57022
[33] Martin Scharlemann and Abigail Thompson, Thin position for 3-manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231 – 238. · Zbl 0818.57013
[34] Jennifer Schultens, The classification of Heegaard splittings for (compact orientable surface)\times \?\textonesuperior , Proc. London Math. Soc. (3) 67 (1993), no. 2, 425 – 448. · Zbl 0789.57012
[35] Jennifer Schultens and Richard Weidman, On the geometric and the algebraic rank of graph manifolds, Pacific J. Math. 231 (2007), no. 2, 481 – 510. · Zbl 1171.57020
[36] Peter B. Shalen, Hyperbolic volume, Heegaard genus and ranks of groups, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 335 – 349. · Zbl 1140.57009
[37] Juan Souto, The rank of the fundamental group of certain hyperbolic 3-manifolds fibering over the circle, The Zieschang Gedenkschrift, Geom. Topol. Monogr., vol. 14, Geom. Topol. Publ., Coventry, 2008, pp. 505 – 518. · Zbl 1144.57017
[38] William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357 – 381. · Zbl 0496.57005
[39] Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56 – 88. · Zbl 0157.30603
[40] Friedhelm Waldhausen, Some problems on 3-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 313 – 322. · Zbl 0414.18010
[41] J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. (2) 37 (1936), no. 4, 782 – 800. · Zbl 0015.24804
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.