Abstract
We give a proof of the so-called generalized Waldhausen conjecture, which says that an orientable irreducible atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy. Jaco and Rubinstein have announced a proof of this conjecture using different methods.
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Agol, I., Li, T.: An algorithm to detect laminar 3-manifolds. Geom. Topol. 7, 287–309 (2003)
Boileau, M., Collins, D.J., Zieschang, H.: Genus 2 Heegaard decompositions of small Seifert manifolds. Ann. Inst. Fourier 41, 1005–1024 (1991)
Bonahon, F., Otal, J.P.: Scindements de Heegaard des espaces lenticulaires. Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 451–466 (1983)
Camacho, C., Neto, A.L.: Geometric theory of foliations. Translated by Sue Goodman. Boston: Birkhäuser 1985
Candel, A., Conlon, L.: Foliations I. Graduate Studies in Math., vol. 23. Providence, RI: American Mathematical Society 2000
Candel, A., Conlon, L.: Foliations II. Graduate Studies in Math., vol. 60. Providence, RI: American Mathematical Society 2003
Casson, A., Gordon, C.: Reducing Heegaard splittings. Topol. Appl. 27, 275–283 (1987)
Floyd, W., Oertel, U.: Incompressible surfaces via branched surfaces. Topology 23, 117–125 (1984)
Gabai, D.: Foliations and 3-manifolds. Proceedings of the International Congress of Mathematicians, Kyoto, 1990, vol. I, II, pp. 609–619. Tokyo: Math. Soc. Japan 1991
Gabai, D.: Taut foliations of 3-manifolds and suspensions of S 1. Ann. Inst. Fourier 42, 193–208 (1992)
Gabai, D.: Essential laminations and Kneser normal form. J. Differ. Geom. 53, 517–574 (1999)
Gabai, D., Oertel, U.: Essential laminations in 3-manifolds. Ann. Math. (2) 130, 41–73 (1989)
Haken, W.: Theorie der Normalflächen: Ein Isotopiekriterium für der Kreisknoten. Acta Math. 105, 245–375 (1961)
Haken, W.: Some results on surfaces in 3-manifolds. In: Studies in Modern Topology, Math. Assoc. Amer. Studies in Math., vol. 5, pp. 39–98. Englewood Cliffs, N.J.: Prentice-Hall 1968
Hatcher, A.: Measured lamination spaces for surfaces, from the topological viewpoint. Topol. Appl. 30, 63–88 (1988)
Jaco, W., Rubinstein, H.: 0-efficient triangulations of 3-manifolds. J. Differ. Geom. 65, 61–168 (2003)
Johannson, K.: Heegaard surfaces in Haken 3-manifolds. Bull. Am. Math. Soc. 23, 91–98 (1990)
Johannson, K.: Topology and combinatorics of 3-manifolds. Lecture Notes in Mathematics, vol. 1599. Berlin: Springer 1995
King, S.: Almost normal Heegaard surfaces. arXiv:math.GT/0303377
Kneser, H.: Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresber. Dtsch. Math.-Ver. 38, 248–260 (1929)
Kobayashi, T.: A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth. Osaka J. Math. 29, 653–674 (1992)
Lackenby, M.: The asymptotic behaviour of Heegaard genus. Math. Res. Lett. 11, 139–149 (2004)
Li, T.: Laminar branched surfaces in 3-manifolds. Geom. Topol. 6, 153–194 (2002)
Li, T.: Boundary curves of surfaces with the 4-plane property. Geom. Topol. 6, 609–647 (2002)
Li, T.: An algorithm to find vertical tori in small Seifert fiber spaces. Comment. Math. Helv. 81, 727–753 (2006)
Li, T.: Commutator groups and foliations without holonomy. Proc. Am. Math. Soc. 130, 2471–2477 (2002)
Li, T.: Heegaard surfaces and measured laminations, II: non-Haken 3-manifolds. J. Am. Math. Soc. 19, 625–657 (2006)
Morgan, J., Shalen, P.: Degerations of hyperbolic structures, II: Measured laminations in 3-manifolds. Ann. Math. 127, 403–456 (1988)
Moriah, Y.: Heegaard splittings of Seifert fibered spaces. Invent. Math. 91, 465–481 (1988)
Moriah, Y., Schultens, J.: Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal. Topology 37, 1089–1112 (1998)
Mosher, L., Oertel, U.: Spaces which are not negatively curved. Commun. Anal. Geom. 6, 67–140 (1998)
Mosher, L., Oertel, U.: Two-dimensionsl measured laminations of positive Euler characteristic. Q. J. Math. 52, 195–216 (2001)
Namazi, H.: Big handlebody distance implies finite mapping class group. arcXiv:math.GT/0406551
Novikov, S.P.: Topology of foliations. Mosc. Math. Soc. 14, 268–305 (1963)
Oertel, U.: Measured laminations in 3-manifolds. Trans. Am. Math. Soc. 305, 531–573 (1988)
Rubinstein, H.: Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds. Proc. Georgia Topology Conference. Am. Math. Coc./Intl. Press 1993
Scharlemann, M.: Local detection of strongly irreducible Heegaard splittings. Topol. Appl. 90, 135–147 (1998)
Schleimer, S.: The disjoint curve property. Geom. Topol. 8, 77–113 (2004)
Sedgwich, E.: An infinite collection of Heegaard splittings that are equivalent after one stabilization. Math. Ann. 308, 65–72 (1997)
Solodov, V.V.: Components of topological foliations. Mat. Sb. 119, 340–354 (1982)
Stocking, M.: Almost normal surfaces in 3-manifolds. Trans. Am. Math. Soc. 352, 171–207 (2000)
Tamura, I.: Topology of Foliations: An Introduction. Translations of Mathematical Monographs, vol. 97. Providence, RI: American Mathematical Society 1992
Waldhausen, F.: Heegaard-Zerlegungen der 3-Sphäre. Topology 7, 195–203 (1968)
Waldhausen, F.: Some problems on 3-manifolds. Proc. Symp. Pure Math. 32, 313–322 (1978)
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Li, T. Heegaard surfaces and measured laminations, I: The Waldhausen conjecture. Invent. math. 167, 135–177 (2007). https://doi.org/10.1007/s00222-006-0009-y
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DOI: https://doi.org/10.1007/s00222-006-0009-y