Finite Dehn surgery on knots. (English) Zbl 0936.57010

Boyer, S.; Zhang, X.

J. Am. Math. Soc. 9, No. 4, 1005-1050 (1996).

This paper studies the problem which Dehn surgery on a knot in a closed orientable 3-manifold can produce 3-manifolds wilh finite [resp. cyclic] fundamental group. Such a surgery is called a finite [resp. cyclic] surgery and its surgery slope is called a finite [resp. cyclic] surgery slope. It is shown that if the exterior \(M\) of a knot \(K\) is hyperbolic, then (1) there are at most six finite/cyclic surgeries on \(K\), and the distance between two finite/cyclic surgery slopes of \(K\) is at most 5; (2) the distance between a finite/cyclic surgery slope and a cyclic surgery slope of \(K\) is at most 2. The result (1) sharply improves the bounds obtained by S. A. Bleiler and C. D. Hodgson [Topology 35, No. 3, 809-833 (1996; Zbl 0863.57009)] by using the \(2\pi\) theorem of Gromov and Thurston. Moreover, the authors conjecture that the number 5 in (1) is replaced with 3. Further results of the case when \(K\) is a knot in \(S^3\) are also obtained.
The approach in this paper follows that developed by M. Culler, C. McA. Gordon, J. Luecke and P. B. Shalen [Ann. Math., II. Ser. 125, 237-300 (1987; Zbl 0633.57006)] for analyzing cyclic surgery slopes, and sharp bounds on the Culler-Shalen norm of finite surgery slopes are obtained.
Studies of the case when \(M\) contains an essential torus are also contained. In particular, it is shown that a satellite knot \(K\) in \(S^3\) admits a non-trivial finite cyclic surgery, then \(K\) is a cabled knot over a torus knot: this gives the complete classification of finite surgeries on satellite knots in \(S^3\). The paper also fcontains various examples which show the extent to which the results are sharp.

Reviewer: M.Sakuma (Osaka)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57R65	Surgery and handlebodies
57N10	Topology of general \(3\)-manifolds (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

exceptional surgery; hyperbolic knot; cyclic surgery; Culler-Shalen norm; finite surgery

Citations:

Zbl 0863.57009; Zbl 0633.57006

References:

[1]	John Berge, The knots in \?²\times \?\textonesuperior which have nontrivial Dehn surgeries that yield \?²\times \?\textonesuperior , Topology Appl. 38 (1991), no. 1, 1 – 19. · Zbl 0725.57001 · doi:10.1016/0166-8641(91)90037-M
[2]	S. Bleiler and C. Hodgson, Spherical space forms and Dehn fillings, preprint. · Zbl 0863.57009
[3]	S. Boyer and A. Nicas, Varieties of group representations and Casson’s invariant for rational homology 3-spheres, Trans. Amer. Math. Soc. 322 (1990), no. 2, 507 – 522. · Zbl 0707.57009
[4]	Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. · Zbl 0568.57001
[5]	Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237 – 300. · Zbl 0633.57006 · doi:10.2307/1971311
[6]	Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109 – 146. · Zbl 0529.57005 · doi:10.2307/2006973
[7]	Ronald Fintushel and Ronald J. Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980), no. 1, 33 – 51. , https://doi.org/10.1007/BF01161380 Ronald Fintushel and Ronald Stern, Correction to: ”Constructing lens spaces by surgery on knots”, Math. Z. 178 (1981), no. 1, 143. · Zbl 0466.57005 · doi:10.1007/BF01218377
[8]	David Gabai, Surgery on knots in solid tori, Topology 28 (1989), no. 1, 1 – 6. · Zbl 0678.57004 · doi:10.1016/0040-9383(89)90028-1
[9]	David Gabai, 1-bridge braids in solid tori, Topology Appl. 37 (1990), no. 3, 221 – 235. · Zbl 0817.57006 · doi:10.1016/0166-8641(90)90021-S
[10]	David Gabai, Foliations and the topology of 3-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461 – 478. David Gabai, Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479 – 536. · Zbl 0627.57012
[11]	William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200 – 225. · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9
[12]	C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687 – 708. · Zbl 0519.57005
[13]	John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. · Zbl 0345.57001
[14]	H. Hopf, Zum Clifford-Kleinschen raumproblem, Math. Ann. 95 (1925-26) 313-319.
[15]	A. Hatcher and U. Oertel, Boundary slopes for Montesinos knots, Topology 28 (1989), no. 4, 453 – 480. · Zbl 0686.57006 · doi:10.1016/0040-9383(89)90005-0
[16]	William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. · Zbl 0433.57001
[17]	Dennis Johnson and John J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 48 – 106. · Zbl 0664.53023 · doi:10.1007/978-1-4899-6664-3_3
[18]	Noriko Maruyama, On Dehn surgery along a certain family of knots, J. Tsuda College 19 (1987), 261 – 280.
[19]	S. V. Matveev and A. T. Fomenko, Isoenergetic surfaces of Hamiltonian systems, the enumeration of three-dimensional manifolds in order of growth of their complexity, and the calculation of the volumes of closed hyperbolic manifolds, Uspekhi Mat. Nauk 43 (1988), no. 1(259), 5 – 22, 247 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 1, 3 – 24. · Zbl 0671.58008 · doi:10.1070/RM1988v043n01ABEH001554
[20]	J. Milnor, Groups which act on \(S^n\) without fixed points, Amer. J. Math. 79 (1957) 623-631. · Zbl 0078.16304
[21]	Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737 – 745. · Zbl 0202.54701
[22]	José M. Montesinos, Surgery on links and double branched covers of \?³, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Princeton Univ. Press, Princeton, N.J., 1975, pp. 227 – 259. Ann. of Math. Studies, No. 84.
[23]	David Mumford, Algebraic geometry. I, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. · Zbl 0356.14002
[24]	Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295 – 323. · Zbl 0103.01802 · doi:10.2307/1970335
[25]	Martin Scharlemann, Producing reducible 3-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481 – 500. · Zbl 0727.57015 · doi:10.1016/0040-9383(90)90017-E
[26]	Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. · Zbl 0355.20006
[27]	I. R. Shafarevich, Basic algebraic geometry, Springer Study Edition, Springer-Verlag, Berlin-New York, 1977. Translated from the Russian by K. A. Hirsch; Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. · Zbl 0284.14001
[28]	D. Tanguay, Chirurgie finie et noeuds rationnels, doctoral dissertation, UQAM, 1995.
[29]	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
[30]	William P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203 – 246. · Zbl 0668.57015 · doi:10.2307/1971277
[31]	——, The geometry and topology of 3-manifolds, Lecture notes, Princeton University, 1977.
[32]	J. Weeks, Ph.D. thesis, Princeton University 1985.
[33]	André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149 – 157. · Zbl 0192.12802 · doi:10.2307/1970495
[34]	Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. · Zbl 0162.53304
[35]	X. Zhang, On property I for knots in \(S^3\), Trans. Amer. Math. Soc., to appear. · Zbl 0797.57006