Smooth $2$-knots in $S^ 2\times S^ 2$ with simply-connected complements are topologically unique
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- by Yoshihisa Sato PDF
- Proc. Amer. Math. Soc. 105 (1989), 479-485 Request permission
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
- Francis Bonahon, Difféotopies des espaces lenticulaires, Topology 22 (1983), no. 3, 305–314 (French). MR 710104, DOI 10.1016/0040-9383(83)90016-2
- Steven Boyer, Simply-connected $4$-manifolds with a given boundary, Trans. Amer. Math. Soc. 298 (1986), no. 1, 331–357. MR 857447, DOI 10.1090/S0002-9947-1986-0857447-6
- Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
- Herman Gluck, The embedding of two-spheres in the four-sphere, Trans. Amer. Math. Soc. 104 (1962), 308–333. MR 146807, DOI 10.1090/S0002-9947-1962-0146807-0 C. D. Hodgson, Involutions and isotopies of lens spaces, Master’s Thesis, Univ. of Melbourne, 1981.
- Rob Kirby (ed.), $4$-manifold problems, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 513–528. MR 780598, DOI 10.1090/conm/035/780598
- Ken’ichi Kuga, Representing homology classes of $S^{2}\times S^{2}$, Topology 23 (1984), no. 2, 133–137. MR 744845, DOI 10.1016/0040-9383(84)90034-X
- Youn W. Lee, Contractibly embedded $2$-spheres in $S^{2}\times S^{2}$, Proc. Amer. Math. Soc. 85 (1982), no. 2, 280–282. MR 652458, DOI 10.1090/S0002-9939-1982-0652458-X
- Richard Mandelbaum, Four-dimensional topology: an introduction, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 1–159. MR 551752, DOI 10.1090/S0273-0979-1980-14687-X
- Frank Quinn, Isotopy of $4$-manifolds, J. Differential Geom. 24 (1986), no. 3, 343–372. MR 868975
- E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. MR 195085, DOI 10.1090/S0002-9947-1965-0195085-8
Additional Information
- © Copyright 1989 American Mathematical Society
- 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