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Dehn surgery and satellite knots


Author: C. McA. Gordon
Journal: Trans. Amer. Math. Soc. 275 (1983), 687-708
MSC: Primary 57M25; Secondary 57N10
DOI: https://doi.org/10.1090/S0002-9947-1983-0682725-0
MathSciNet review: 682725
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Abstract: For certain kinds of $ 3$-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in $ {S^3}$ is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain $ {S^3}$, a fake $ {S^3}$, a fake $ {S^3}$ with nonzero Rohlin invariant, $ {S^1} \times {S^2}$, a fake $ {S^1} \times {S^2}, {S^1} \times {S^2} \char93 M$ with $ M$ nonsimply-connected, or a fake lens space.


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  • [1] R. H. Bing and J. M. Martin, Cubes with knotted holes, Trans. Amer. Math. Soc. 155 (1971), 217-331. MR 0278287 (43:4018a)
  • [2] W. R. Brakes, Property $ R$ and superslices, Quart. J. Math. Oxford 31 (1980), 263-281. MR 587091 (83e:57006)
  • [3] R. Fintushel and R. J. Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980), 33-51. MR 595630 (82i:57009a)
  • [4] M. Gerstenhaber and O. S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531-1533. MR 0166296 (29:3573)
  • [5] F. González-Acuña, Dehn's construction on knots, Bol. Soc. Mat. Mexicana 15 (1970), 58-79. MR 0356022 (50:8495)
  • [6] C. McA. Gordon, Knots, homology spheres, and contractible $ 4$-manifolds, Topology 14 (1975), 151-172. MR 0402762 (53:6576)
  • [7] C. McA. Gordon and D. W. Sumners, Knotted ball pairs whose product with an interval is unknotted, Math. Ann. 217 (1975), 47-52. MR 0380816 (52:1713)
  • [8] K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. 7 (1957), 29-62. MR 0087652 (19:386a)
  • [9] A. Hatcher, On the boundary curves of incompressible surfaces (to appear). MR 658066 (83h:57016)
  • [10] J. Hempel, $ 3$-Manifolds, Ann. of Math. Studies, No. 86, Princeton Univ. Press, Princeton, N.J., 1976. MR 0415619 (54:3702)
  • [11] W. H. Jaco and P. B. Shalen, Seifert fibred spaces in $ 3$-manifolds, Mem. Amer. Math. Soc. 21 (1979). MR 539411 (81c:57010)
  • [12] K. Johannson, Homotopy equivalences of $ 3$-manifolds with boundaries, Lecture Notes in Math., vol. 761, Springer-Verlag, Berlin, Heidelberg and New York, 1979. MR 551744 (82c:57005)
  • [13] M. A. Kervaire, On higher dimensional knots, Differential and Combinatorial Topology, A Symposium in Honor of Marston Morse (S. S. Cairns, Editor), Princeton Univ. Press, Princeton, N.J., 1965, pp. 105-119. MR 0178475 (31:2732)
  • [14] R. Kirby, Problems in low dimensional manifold theory, Proc. Amer. Math. Soc. Summer Institute in Topology, Stanford, (1976), Amer. Math. Soc., Providence, R.I., Proc. Sympos. Pure Math., Vol. 32, 1978, pp. 273-312. MR 520548 (80g:57002)
  • [15] R. Kirby and P. Melvin, Slice knots and Property $ R$, Invent. Math. 45 (1978), 57-59. MR 0467754 (57:7606)
  • [16] R. A. Litherland, Surgery on knots in solid tori, Proc. London Math. Soc. 39 (1979), 130-146. MR 538135 (80i:57005)
  • [17] -, Surgery on knots in solid tori. II, J. London Math. Soc. 22 (1980), 559-569. MR 596334 (82d:57002)
  • [18] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Ergebnisse der Math. und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin, Heidelberg and New York, 1977. MR 0577064 (58:28182)
  • [19] J. W. Morgan, Morse-Smale flows on $ 3$-dimensional manifolds, Topology 18 (1979), 41-53. MR 528235 (80j:58043)
  • [20] L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737-745. MR 0383406 (52:4287)
  • [21] D. Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)
  • [22] H. Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131-286. MR 0072482 (17:291d)
  • [23] H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta. Math. 60 (1933), 147-238. MR 1555366
  • [24] J. L. Shaneson, Embeddings with codimension two of spheres in spheres and $ H$-cobordisms of $ {S^1} \times {S^3}$, Bull. Amer. Math. Soc. 74 (1968), 972-974. MR 0230325 (37:5887)
  • [25] L. Siebenmann, On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology $ 3$-spheres, Proc. Topology Sympos., Siegen, 1979 (U. Koschorke and W. D. Neumann, Editors), Lecture Notes in Math., vol. 788, Springer-Verlag, Berlin, Heidelberg and New York, 1980. MR 585660 (81k:57011)
  • [26] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. I (to appear). MR 956596 (89k:57023)
  • [27] -, The geometry and topology of $ 3$-manifolds, preprints, Princeton University.
  • [28] F. Waldhausen, Eine Klasse von $ 3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308-333; 4 (1967), 87-117. MR 0235576 (38:3880)
  • [29] J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409-428. MR 0004123 (2:323c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0682725-0
Keywords: Dehn surgery, satellite knots, simple knots
Article copyright: © Copyright 1983 American Mathematical Society

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