Dehn surgery and satellite knots

Gordon, C. McA.

doi:10.1090/S0002-9947-1983-0682725-0

Dehn surgery and satellite knots
HTML articles powered by AMS MathViewer

by C. McA. Gordon PDF

Trans. Amer. Math. Soc. 275 (1983), 687-708 Request permission

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 ${S^1} \times {S^2}, {S^1} \times {S^2} \# M$ with $M$ nonsimply-connected, or a fake lens space.

References

R. H. Bing and J. M. Martin, Cubes with knotted holes, Trans. Amer. Math. Soc. 155 (1971), 217–231. MR 278287, DOI 10.1090/S0002-9947-1971-0278287-4
W. R. Brakes, Property $\textrm {R}$ and superslices, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 123, 263–281. MR 587091, DOI 10.1093/qmath/31.3.263
Ronald Fintushel and Ronald J. Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980), no. 1, 33–51. MR 595630, DOI 10.1007/BF01161380
Murray Gerstenhaber and Oscar S. Rothaus, The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1531–1533. MR 166296, DOI 10.1073/pnas.48.9.1531
F. González-Acuña, Dehn’s construction on knots, Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79. MR 356022
C. McA. Gordon, Knots, homology spheres, and contractible $4$-manifolds, Topology 14 (1975), 151–172. MR 402762, DOI 10.1016/0040-9383(75)90024-5
C. McA. Gordon and D. W. Sumners, Knotted ball pairs whose product with an interval is unknotted, Math. Ann. 217 (1975), no. 1, 47–52. MR 380816, DOI 10.1007/BF01363239
K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR 87652, DOI 10.1112/plms/s3-7.1.29
A. E. Hatcher, On the boundary curves of incompressible surfaces, Pacific J. Math. 99 (1982), no. 2, 373–377. MR 658066
John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
Michel A. Kervaire, On higher dimensional knots, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 105–119. MR 0178475
Rob Kirby, Problems in low dimensional manifold theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 273–312. MR 520548
Robion Kirby and Paul Melvin, Slice knots and property $\textrm {R}$, Invent. Math. 45 (1978), no. 1, 57–59. MR 467754, DOI 10.1007/BF01406223
R. A. Litherland, Surgery on knots in solid tori, Proc. London Math. Soc. (3) 39 (1979), no. 1, 130–146. MR 538135, DOI 10.1112/plms/s3-39.1.130
R. A. Litherland, Surgery on knots in solid tori. II, J. London Math. Soc. (2) 22 (1980), no. 3, 559–569. MR 596334, DOI 10.1112/jlms/s2-22.3.559
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
John W. Morgan, Nonsingular Morse-Smale flows on $3$-dimensional manifolds, Topology 18 (1979), no. 1, 41–53. MR 528235, DOI 10.1016/0040-9383(79)90013-2
Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745. MR 383406
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Horst Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131–286 (German). MR 72482, DOI 10.1007/BF02392437
H. Seifert, Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238 (German). MR 1555366, DOI 10.1007/BF02398271
Julius 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 230325, DOI 10.1090/S0002-9904-1968-12107-X
L. Siebenmann, On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology $3$-spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 172–222. MR 585660
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6

The geometry and topology of

manifolds

Friedhelm Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117 (German). MR 235576, DOI 10.1007/BF01402956
J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428. MR 4123, DOI 10.2307/1968907