Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Smooth $ 2$-knots in $ S\sp 2\times S\sp 2$ with simply-connected complements are topologically unique


Author: Yoshihisa Sato
Journal: Proc. Amer. Math. Soc. 105 (1989), 479-485
MSC: Primary 57Q45; Secondary 57N13, 57R40
DOI: https://doi.org/10.1090/S0002-9939-1989-0940880-X
MathSciNet review: 940880
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a given primitive homology class $ \xi $ of $ {H_2}({S^2} \times {S^2};{\mathbf{Z}})$, we show that there exists only one smoothly embedded $ 2$-sphere in $ {S^2} \times {S^2}$ , up to homeomorphism, which represents $ \xi $ and whose complement is simply connected.


References [Enhancements On Off] (What's this?)

  • [1] F. Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983), 305-314. MR 710104 (85d:57008)
  • [2] S. Boyer, Simply-connected $ 4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), 331-357. MR 857447 (88b:57023)
  • [3] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. MR 679066 (84b:57006)
  • [4] H. Gluck, The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308-333. MR 0146807 (26:4327)
  • [5] C. D. Hodgson, Involutions and isotopies of lens spaces, Master's Thesis, Univ. of Melbourne, 1981.
  • [6] R. Kirby, $ 4$-manifold problems, Contemp. Math. 35 (1984), 513-528. MR 780598 (86f:57016)
  • [7] K. Kuga, Representing homology classes of $ {S^2} \times {S^2}$, Topology 23 (1984), 133-137. MR 744845 (85m:57011)
  • [8] Y. W. Lee, Contractibly embedded $ 2$-spheres in $ {S^2} \times {S^2}$, Proc. Amer. Math. Soc. 85 (1982), 280-282. MR 652458 (84c:57012)
  • [9] R. Mandelbaum, Four-dimensional topology: an introduction, Bull. Amer. Math. Soc. 2 (1980), 1-159. MR 551752 (81j:57001)
  • [10] F. Quinn, Isotopy of $ 4$-manifolds, J. Differential Geom. 24 (1986), 343-372. MR 868975 (88f:57020)
  • [11] E. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495. MR 0195085 (33:3290)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57Q45, 57N13, 57R40

Retrieve articles in all journals with MSC: 57Q45, 57N13, 57R40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0940880-X
Keywords: $ 2$-knot, $ {S^2} \times {S^2}$
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society