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Dehn filling: A survey

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In this paper we give a brief survey of the present state of knowledge on exceptional Dehn fillings on 3-manifolds with torus boundary. For our discussion, it is necessary to first give a quick overview of what is presently known, and what is conjectured, about the structure of 3-manifolds. This is done in Section 2. In Section 3 we summarize the known bounds on the distances between various kinds of exceptional Dehn fillings, and compare these with the distances that arise in known examples. In Section 4 we make some remarks on the special case of complements of knots in the 3-sphere. We have chosen to phrase questions as conjectures; this gives them a certain edge and perhaps increases the likelihood that someone will try to (dis)prove them. Incidentally, no particular claim is made for unattributed conjectures; most of them are lore to the appropriate folk. Related survey articles are [Go1] and [Lu]. I would like to thank Pat Callahan, Craig Hodgson, John Luecke, Alan Reid and Eric Sedgwick for helpful conversations, and the referee for his useful comments.
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  • Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712-1082, U.S.A.
  • [A1] C. Adams, The noncompact hyperbolic 3-manifold of minimal volume, Proc. Amer. Math. Soc., 100 (1987), 601-606
  • [A2] C. Adams, Unknotting tunnels in hyperbolic 3-manifolds, Math. Ann., 302 (1995), 177-195
  • [B1] J. Berge, The knots in $D^2 × S^1$ which have nontrivial Dehn surgeries that yield $D^2 × S^1$, Topology and its Applications, 39 (1991), 1-19
  • [B2] J. Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript.
  • [BPZ] S. Betley, J. H. Przytycki and T. Zukowski, Hyperbolic structures on Dehn filling of some punctured-torus bundles over $S^1$, Kobe J. Math, 3 (1986), 117-147
  • [BM] R. H. Bing and J. M. Martin, Cubes with knotted holes, Trans. Amer. Math. Soc., 155 (1971), 217-231
  • [BH1] S. Bleiler and C. Hodgson, Spherical space forms and Dehn surgery, Knots 90, Proceedings of the International Conference on Knot Theory and Related Topics (A. Kawauchi, ed.), Osaka (Japan), de Gruyter, Berlin, New York, 1992, 425-433
  • [BH2] S. Bleiler and C. Hodgson, Spherical space forms and Dehn filling, Topology 35 (1996), 809-833.
  • [BZ1] S. Boyer and X. Zhang, Finite Dehn surgery on knots, preprint.
  • [BZ2] S. Boyer and X. Zhang, The semi-norm and Dehn filling, preprint.
  • [BW] M. Brittenham and Y.-Q. Wu, The classification of Dehn surgeries on 2-bridge knots, preprint.
  • [BFLW] A. M. Brunner, M. L. Frame, Y. W. Lee, and N. J. Wielenberg, Classifying torsion-free subgroups of the Picard group, Trans. Amer. Math. Soc., 282 (1984), 205-235
  • [CHW] P. J. Callahan, M. V. Hildebrand and J. R. Weeks, A census of cusped hyperbolic 3-manifolds, preprint.
  • [CJ] A. Casson and D. Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math., 118 (1994), 441-456
  • [CGLS] M. Culler, C. McA. Gordon, J. Luecke and P. B. Shalen, Dehn surgery on knots, Ann. of Math., 125 (1987), 237-300
  • [D] J. Dean, Ph.D. thesis, University of Texas at Austin, 1996.
  • [Ep] D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math., 95 (1972), 66-82
  • [Eu1] M. Eudave-Muñoz, Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots, Trans. Amer. Math. Soc., 330 (1992), 463-501
  • [Eu2] M. Eudave-Muñoz, Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Proceedings of the Georgia International Topology Conference (1993) (to appear).
  • [Eu3] M. Eudave-Muñoz, 4-punctured tori on the exterior of knots, preprint.
  • [EM] B. Evans and J. Maxwell, Quaternion actions on $S^3$, Amer. J. Math., 101 (1979), 1123-1130
  • [F1] C. D. Feustel, On the torus theorem and its applications, Trans. Amer. Math. Soc., 217 (1976), 1-43
  • [F2] C. D. Feustel, On the torus theorem for closed 3-manifolds, Trans. Amer. Math. Soc., 217 (1976), 45-57
  • [FS] R. Fintushel and R. Stern, Constructing lens spaces by surgery on knots, Math. Z., 175 (1980), 33-51
  • [Ga1] D. Gabai, Foliations and the topology of 3-manifolds, III, J. Diff. Geom., 26 (1987), 479-536
  • [Ga2] D. Gabai, Surgery on knots in solid tori, Topology, 28 (1989), 1-6
  • [Ga3] D. Gabai, Convergence groups are Fuchsian groups, Ann. of Math., 136 (1992), 447-510
  • [Ga4] D. Gabai, Eight problems in the geometric theory of foliations and laminations on 3-manifolds, preprint.
  • [GS] F. González-Acuña and H. Short, Knot surgery and primeness, Math. Proc. Camb. Phil. Soc., 99 (1986), 89-102
  • [Go1] C. McA. Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematicians, Kyoto, 1990, Springer-Verlag, Tokyo, 1991, pp. 631-642
  • [Go2] C. McA. Gordon, Boundary slopes of punctured tori in 3-manifolds, Trans. Amer. Math. Soc. (to appear).
  • [GLi] C. McA. Gordon and R. A. Litherland, Incompressible planar surfaces in 3-manifolds, Topology and its Applications, 18 (1984), 121-144
  • [GLu1] C. McA. Gordon and J. Luecke, Only integral Dehn surgeries can yield reducible manifolds, Math. Proc. Camb. Phil. Soc., 102 (1987), 94-101
  • [GLu2] C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc., 2 (1989), 371-415
  • [GLu3] C. McA. Gordon and J. Luecke, Reducible manifolds and Dehn surgery, Topology 35 (1996), 385-409.
  • [GLu4] C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, I, Communications in Analysis and Geometry, 3 (1995), 597-644
  • [GLu5] C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, II, Comm. Anal. Geom. (to appear).
  • [HS] C. Hayashi and K. Shimokawa, Symmetric knots satisfy the cabling conjecture, preprint.
  • [He] W. Heil, Elementary surgery on Seifert fiber spaces, Yokohama Math. J., 22 (1974), 135-139
  • [HMW] C. D. Hodgson, G. R. Meyerhoff and J. R. Weeks, Surgeries on the Whitehead link yield geometrically similar manifolds, Topology '90 (B. Apanasov, W.D. Neumann, A.W. Reid and L. Siebenmann, eds.) de Gruyter, Berlin, 1992, pp. 195-206.
  • [HW] C. D. Hodgson and J. R. Weeks, A census of closed hyperbolic 3-manifolds, in preparation.
  • [Hof] J. Hoffman, Ph.D. thesis, The University of Texas at Austin, 1995.
  • [Hop] H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann., 95 (1925), 313-319
  • [K] H. Kneser, Geschlossene Flächen in dreidimensionale Mannigfaltigkeiten, Jahresber. Deutsch. Math.-Verein., 38 (1929), 248-260
  • [Le] R. Lee, Semicharacteristic classes, Topology, 12 (1973), 183-199
  • [Li] G. R. Livesay, Fixed point free involutions on the 3-sphere, Ann. of Math., 72 (1960), 603-611
  • [Lu] J. Luecke, Dehn surgery on knots in the 3-sphere, Proceedings of the International Congress of Mathematicians, Zürich, 1994, Birkhäuser Verlag, Switzerland, 1995, pp. 585-594.
  • [MT] W. Menasco and M. Thistlethwaite, Surfaces with boundary in alternating knot exteriors, J. Reine Angew. Math., 426 (1992), 47-65
  • [Me] G. Mess, Centers of 3-manifold groups and groups which are coarse quasiisometric to planes, preprint.
  • [Mi1] J. Milnor, Groups which act on $S^n$ without fixed points, Amer. J. Math., 79 (1957), 623-630
  • [Mi2] J. Milnor, A unique factorization theorem for 3-manifolds, Amer. J. Math., 84 (1962), 1-7
  • [MR] Y. Moriah and H. Rubinstein, Heegaard structures of negatively curved 3-manifolds, preprint.
  • [My] R. Myers, Free involutions on lens spaces, Topology, 20 (1981), 313-318
  • [NR] W. D. Neumann and A. W. Reid, Arithmetic of hyperbolic manifolds, Topology '90, (B. Apanasov, W. D. Neumann, A. W. Reid and L. Siebenmann, eds.), de Gruyter, Berlin, 1992, pp. 273-310.
  • [Oh1] S. Oh, Reducible and toroidal 3-manifolds obtained by Dehn filling, Topology and its Applications, (to appear).
  • [Oh2] S. Oh, Dehn filling, reducible 3-manifolds, and Klein bottles, preprint.
  • [Or] P. Orlik, Seifert manifolds, Lecture Notes in Mathematics, vol. 291, Springer, Berlin, 1972.
  • [Pap] C. D. Papakyriakopoulos, On Dehn's lemma and the asphericity of knots, Ann. of Math., 66 (1957), 1-26
  • [Pat] R. Patton, Incompressible punctured tori in the complements of alternating knots, Math. Ann., 301 (1995), 1-22
  • [Ric] P. M. Rice, Free actions of $Z_4$ on $S^3$, Duke Math. J., 36 (1969), 749-751
  • [Rit] G. X. Ritter, Free $Z_8$ actions on $S^3$, Trans. Amer. Math. Soc., 181 (1973), 195-212
  • [Ru] J. H. Rubinstein, Free actions of some finite groups on $S^3$. I, Math. Ann., 240 (1979), 165-175
  • [Sch1] M. Scharlemann, Sutured manifolds and generalized Thurston norms, J. Diff. Geom., 29 (1989), 557-614
  • [Sch2] M. Scharlemann, Producing reducible 3-manifolds by surgery on a knot, Topology, 29 (1990), 481-500
  • [Sco1] P. Scott, A new proof of the annulus and torus theorems, Amer. J. Math., 102 (1980), 241-277
  • [Sco2] P. Scott, There are no fake Seifert fibre spaces with infinite $π_1$, Ann. of Math., 117 (1983), 35-70
  • [Sco3] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487
  • [Se] H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math., 60 (1932), 147-238
  • [TS] W. Threlfall and H. Seifert, Topologische Untersuchung der Discontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, I, Math. Ann., 104 (1930), 1-70; II, 107 (1932), 543-586
  • [T1] W. P. Thurston, The Geometry and Topology of 3-manifolds, Princeton University, 1978.
  • [T2] W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6 (1982), 357-381
  • [T3] W. P. Thurston, Hyperbolic structures on 3-manifolds, I: Deformation of acylindrical manifolds, Ann. of Math., 124 (1986), 203-246
  • [Wa] F. Waldhausen, On the determination of some bounded 3-manifolds by their fundamental groups alone, Proc. Inter. Sym. Topology, Hercy-Novi, Yugoslavia, 1968; Beograd, 1969, pp. 331-332.
  • [WS] C. Weber and H. Seifert, Die beiden Dodekaederräume, Math. Z., 37 (1933), 237-253
  • [We] J. R. Weeks, Hyperbolic structures on three-manifolds, Ph.D. thesis, Princeton University, 1985.
  • [Wh] J. H. C. Whitehead, On 2-spheres in 3-manifolds, Bull. Amer. Math. Soc., 64 (1958), 161-166
  • [Wu1] Y.-Q. Wu, Dehn surgery on arborescent knots, J. Diff. Geom. 43 (1996), 171-197.
  • [Wu2] Y.-Q. Wu, Dehn fillings producing reducible manifold and toroidal manifold, preprint.
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