[1] Agol I. The virtual Haken conjecture. With an appendix by Ian Agol, Daniel Groves, and Jason Manning. Doc Math, 2013, 18: 1045-1087
[2] Bachman D. Connected sums of unstabilized Heegaard splittings are unstabilized. Geom Topol, 2008, 12(4): 2327-2378
[3] Bachman D. Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds. Math Ann, 2013, 355(2): 697-728
[4] Berge J. Some knots with surgeries yielding lens spaces. Unpublished Manuscript, 1990
[5] Bing R H. An alternative proof that 3-manifolds can be triangulated. Ann of Math, 1959, 69: 37-65
[6] Boileau M, Otal J -P. Scindements de Heegaard des variétés de Seifert elliptiques et euclidiennes. (French) J London Math Soc, 2001, 63(2): 501-512
[7] Bonahon F, Otal J -P. Scindements de Heegaard des espaces lenticulaires. (French) Ann Sci École Norm Sup (4), 1983, 16(3): 451-466
[8] Boileau M, Otal J -P. Sur les scindements de Heegaard du tore T3. (French) [On the Heegaard splittings of the torus T3]. J Differential Geom, 1990, 32(1): 209-233
[9] Boileau M, Zieschang H. Heegaard genus of closed orientable Seifert 3-manifolds. Invent Math, 1984, 76(3): 455-468
[10] Casson A J, Gordon C. Reducing Heegaard splittings. Topol Appl, 1987, 27: 275-283
[11] Colding T H, Gabai D, Ketover D. On the classification of Heegaard splittings. Duke Math J, 2018, 167(15): 2833-2856
[12] Culler M, Gordon C M, Luecke J, Shalen P B. Dehn surgery on knots. Ann of Math, 1987, 125(2): 237-300
[13] Cho S, Koda Y. The mapping class groups of reducible Heegaard splittings of genus two. Trans Amer Math Soc, 2019, 371(4): 2473-2502
[14] Du K, Gao X. A note on Heegaard splittings of amalgamated 3-manifolds. Chinese Ann Math Ser B, 2011, 32(3): 475-482
[15] Gordon C M, Luecke J. Links with unlinking number one are prime. Proc Amer Math Soc, 1994, 120(4): 1271-1274
[16] Gordon C M. Dehn filling a survey//Knot Theory, Warsaw, 1995. Warsaw: Polish Acad Sci, 1998: 129-144
[17] Gordon C M, Moriah Y. Workshop on Heegaard Splittings. Proceedings of the workshop held in Haifa, July 10-20, 2005. Geometry & Topology Monographs, 12. Coventry: Geometry and Topo Logy Publisher, 2007
[18] Greene J. The lens space realization problem. Annals of Mathematics, 2013, 177(2): 449-511
[19] Haken W. Some results on surfaces in 3-manifolds//Studies in Modern Topology. Math Assoc Amer, 1968: 39-98
[20] Harvey W J. Boundary structure of the modular group, Riemann surfaces and related topics//Kra I, Maskit B, eds. Proceedings of the 1978 Stony Brook Conference. Ann of Math Stud Vol 97. Princeton: Princeton Univ Press, 1981: 245-251
[21] Hartshorn K. Heegaard splittings of Haken manifolds have bounded distance. Pacific J Math, 2002, 204: 61-75
[22] Heegaard P. Forstudiertil en topologisk Teori for de algebraiske Fladers Sammenhang. Copenhagen: Copenhagen University, 1898
[23] Hempel J. 3-manifolds as viewed from the curve complex. Topology, 2001, 40(3): 631-657
[24] Hass J, Thompson A, Thurston W. Stabilization of Heegaard splittings. Geom Topol, 2009, 13(4): 2029- 2050
[25] Ido A, Jang Y, Kobayashi T. Heegaard splittings of distance exactly n. Algebr Geom Topol, 2014, 14(3): 1395-1411
[26] Johannson K. Heegaard surfaces in Haken 3-manifolds. Bull Amer Math Soc (NS), 1990, 23(1): 91-98
[27] Johannson K. Topology and Combinatorics of 3-manifolds. Lecture Notes in Mathematics, 1599. Berlin: Springer-Verlag, 1995
[28] Johnson J. Stable functions and common stabilizations of Heegaard splittings. Trans Amer Math Soc, 2009, 361(7): 3747-3765
[29] Johnson J. Bounding the stable genera of Heegaard splittings from below. J Topol, 2010, 3(3): 668-690
[30] Johnson J. Mapping class groups of medium distance Heegaard splittings. Proc Amer Math Soc, 2010, 138(12): 4529-4535
[31] Kneser H. Geschlossen Flachen in dreidimensionalen Mannigfaltigkeiten. Jcahresbericht der Deutschen Mathematiker Vereinigung, 1929, 38: 248-260
[32] Kirby R. Problems in low-dimensional topology//Geometric Topology, AMS/IP Studies in Advanced Mathematics, Vol 2. Cambridge, MA: International Press, 1997: 35-473
[33] Kobayashi T. Casson-Gordon’s rectangle condition of Heegaard diagrams and incompressible tori in 3- manifolds. Osaka J Math, 1988, 25(3): 553-573
[34] Kobayashi T. A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum. J Knot Theory Ramifications, 1994, 3(2): 179-186
[35] Kobayashi T, Rieck Y. Knot exteriors with additive Heegaard genus and Morimoto’s conjecture. Algebr Geom Topol, 2008, 8(2): 953-969
[36] Kobayashi T, Qiu R. The amalgamation of high distance Heegaard splittings is always efficient. Math Ann, 2008, 341(3): 707-715
[37] Kobayashi T. Qiu R, Rieck Y, Wang S C. Separating incompressible surfaces and stabilizations of Heegaard splittings. Math Proc Cambridge Philos Soc, 2004, 137(3): 633-643
[38] Kobayashi T, Saeki O. The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map. Pacific Journal of Mathematics, 2000, 195(1): 101-156
[39] Lackenby M. The Heegaard genus of amalgamated 3-manifolds. Geom Dedicata, 2004, 109: 139-145
[40] Lei F. On stability of Heegaard splittings. Math Proc Cambridge Philos Soc, 2000, 129(1): 55-57
[41] Li T. Heegaard surfaces and measured laminations. II. Non-Haken 3-manifolds. J Amer Math Soc, 2006, 19(3): 625-657
[42] Li T. Heegaard surfaces and measured laminations. I. The Waldhausen conjecture. Invent Math, 2007, 167(1): 135-177
[43] Li T. Heegaard surfaces and the distance of amalgamation. Geom Topol, 2010, 14(4): 1871-1919
[44] Li T. Saddle tangencies and the distance of Heegaard splittings. Algebr Geom Topol, 2007, 7: 1119-1134
[45] Li T. Rank and genus of 3-manifolds. J Amer Math Soc, 2013, 26(3): 777-829
[46] Li T, Qiu R. On the degeneration of tunnel numbers under a connected sum. Trans Amer Math Soc, 2016, 368(4): 2793-2807
[47] Li T, Moriah Y, Pinsky T. Tunnel number one knots satisfy the Berge conjecture. arxiv:1701.01421
[48] Lickorish W B R. A representation of orientable combinatorial 3-manifolds. Ann of Math, 1962, 76(3): 531-540
[49] Masur H A, Minsky Y N. Geometry of the complex of curves. I. Hyperbolicity. Invent Math, 1999, 138(1):103-149
[50] Masur H A. Minsky Y N. Geometry of the complex of curves. II. Hierarchical structure. Geom Funct Anal, 2000, 10(4): 902-974
[51] Minsky Y, Moriah Y, Schleimer S. High distance knots. Algebr Geom Topol, 2007, 7: 1471-1483
[52] Ma J, Qiu R, Zou Y. Non degenerating Dehn fillings on genus two Heegaard splittings of knots’ complements. Sci China Math, 2018, 61(6): 1099-1108
[53] Milnor J. A unique decomposition theorem for 3-manifolds. Amer J Math, 1962, 84: 1-7
[54] Moriah Y, Schultens J. Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal. Topology, 1998, 37(5): 1089-1112
[55] Moriah Y. Heegaard splittings of knot exteriors//Workshop on Heegaard Splittings. Geom Topol Monogr, Vol 12. Coventry: Geom Topol Publ, 2007: 191-232
[56] Moise M E. Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann of Math, 1952, 56: 96-114
[57] Morimoto K. There are knots whose tunnel numbers go down under connected sum. Proc Amer Math Soc, 1995, 123(11): 3527-3532
[58] Morimoto K. On the super additivity of tunnel number of knots. Math Ann, 2000, 317(3): 489-508
[59] Namazi H. Big Heegaard distance implies finite mapping class group. Topology Appl, 2007, 154(16): 2939- 2949
[60] Namazi H, Souto J. Heegaard splittings and pseudo-Anosov maps. Geom Funct Anal, 2009, 19(4): 1195- 1228
[61] Ni Y. Knot Floer homology detects fibred knots. Invent Math, 2007, 170: 577-608
[62] Norwood F H. Every two-generator knot is prime. Proc Amer Math Soc, 1982, 86(1): 143-147
[63] Qiu R, Scharlemann M. A proof of the Gordon conjecture. Adv Math, 2009, 222(6): 2085-2106
[64] Qiu R, Zou Y, Guo Q. The Heegaard distances cover all nonnegative integers. Pacific J Math, 2015, 275(1): 231-255
[65] Qiu R, Zou Y. 3-manifolds admitting locally large distance 2 Heegaard splittings. Comm Anal Geom, 2019, 27(6): 1355-1379
[66] Reidemeister K. Zur dreidimensionalen Topologie. (German) Abh Math Sem Univ Hamburg, 1933, 9(1): 189-194
[67] Rubinstein H, Scharlemann M. Comparing Heegaard splittings of non-Haken 3-manifolds. Topology, 1996, 35(4): 1005-1026
[68] Rubinstein H, Scharlemann M. Comparing Heegaard splittings—the bounded case. Trans Amer Math Soc, 1998, 350(2): 689-715
[69] Rubinstein H, Scharlemann M. Genus two Heegaard splittings of orientable three-manifolds//Proceedings of the Kirbyfest (Berkeley, CA, 1998). Geom Topol Monogr, 2. Coventry: Geom Topol Publ, 1999: 489-553
[70] Scharlemann M. Proximity in the curve complex: boundary reduction and bicompressible surfaces. Pacific J Math, 2006, 228(2): 325-348
[71] Scharlemann M, Thompson A. Thin position for 3-manifolds//Geometric topology (Haifa, 1992). Contemp Math, 164. Providence, RI: Amer Math Soc, 1994: 231-238
[72] Scharlemann M, Thompson A. Heegaard splittings of the I-bundle of a surface are standard. Math Ann, 1993, 295(3): 549-564
[73] Scharlemann M, Thompson A. Thin position and Heegaard splittings of the 3-sphere. J Differential Geom, 1994, 39(2): 343-357
[74] Scharlemann M, Schultens J. The tunnel number of the sum of n knots is at least n. Topology, 1999, 38(2): 265-270
[75] Scharlemann M, Schultens J. Annuli in generalized Heegaard splittings and degeneration of tunnel number. Math Ann, 2000, 317(4): 783-820
[76] Scharlemann M, Tomova M. Alternate Heegaard genus bounds distance. Geom Topol, 2006, 10: 593-617
[77] Schirmer T. A lower bound on tunnel number degeneration. Algebr Geom Topol, 2016, 16(3): 1279-1308
[78] Schultens J, Weidmann R. Destabilizing amalgamated Heegaard splittings//Workshop on Heegaard Splittings. Geom Topol Monogr, 12. Coventry: Geom Topol Publ, 2007: 319-334
[79] Schultens J. The classification of Heegaard splittings for (compact orientable surface)×S1. Proc London Math Soc, 1993, 67(2): 425-448
[80] Schultens J. The stabilization problem for Heegaard splittings of Seifert fibered spaces. Topology Appl, 1996, 73(2): 133-139
[81] Schultens J. Additivity of tunnel number for small knots. Comment Math Helv, 2000, 75(3): 353-367
[82] Schultens J, Weidman R. On the geometric and the algebraic rank of graph manifolds. Pacific J Math, 2007, 231(2): 481-510
[83] Sedgewick E. An infinite collection of Heegaard splittings that are equivalent after one stabilization. Math Ann, 1997, 308(1): 65-72
[84] Singer J. Three-dimensional manifolds and their Heegaard diagrams. Trans Amer Math Soc, 1933, 35(1): 88-111
[85] Souto J. The rank of the fundamental group of certain hyperbolic 3-manifolds fibering over the circle//The Zieschang Gedenkschrift, Geom Topol Monogr, Vol 14. Coventry: Geom Topol Publ, 2008: 505-518
[86] Thurston W. The geometry and topology of 3-manifolds. Lecture notes, Princeton University, 1977
[87] Thurston W. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull Amer Math Soc (NS), 1982, 6(3): 357-381
[88] Waldhausen F. On irreducible 3-manifolds which are sufficient large. Ann Math, 1968, 87: 56-88
[89] Waldhausen F. Heegaard-Zerlegungen der 3-Sphäre. Topology, 1968, 7: 195-203
[90] Waldhausen F. Some problems on 3-manifolds//Algebraic and Geometric Topology (Proc Sympos Pure Math, Stanford Univ, Stanford, Calif, 1976), Part 2. Proc Sympos Pure Math, XXXII. Providence, RI: Amer Math Soc, 1978: 313-322
[91] Wallace A H. Modifications and cobounding manifolds. Canadian J Math, 1960, 12: 503-528
[92] Yang G, Lei F. On amalgamations of Heegaard splittings with high distance. Proc Amer Math Soc, 2009, 137(2): 723-731
[93] Zhang F, Qiu R, Zou Y. Infinitely many hyperbolic 3-manifolds admitting distance-d and genus-g Heegaard splittings. Geom Dedicata, 2016, 181: 213-222
[94] Zou Y, Qiu R. Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings. Groups Geom Dyn, 2020, 14(2): 591-605
[95] Zou Y. An upper bound on distance degenerate handle additions. to appear in Commun Anal Geom |