Hostname: page-component-cb9f654ff-k7rjm Total loading time: 0 Render date: 2025-09-01T06:34:58.875Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Asphericity of Knot Complements

Published online by Cambridge University Press:  20 November 2018

Vo Thanh Liem  and
Gerard A. Venema
Vo Thanh Liem
Affiliation:
Princeton University, Princeton, New Jersey 08544, U.S.A.
Gerard A. Venema
Affiliation:
Princeton University, Princeton, New Jersey 08544, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Two examples of topological embeddings of S 2 in S 4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is nontrivial. The second example is a non-locally flat embedding whose complement exhibits this property locally.

Two theorems are proved. The first answers the question of just when good π 1 implies the vanishing of the higher homotopy groups for knot complements in S 4. The second theorem characterizes local flatness for 2-spheres in S 4 in terms of a local π 1 condition.

Keywords

57Q4557N45knot complementfundamental groupasphericity1 –alglocal flatness2–sphere

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 2 , 01 April 1993 , pp. 340 - 356
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Cannon, J., ULC properties in neighborhood of embedded surfaces and curves in E3, Can. J. Math. 25(1973), 31–73.Google Scholar
2. Chapman, T.A., Compact Hilbert cube manifolds and the invariance of Whitehead torsion, Bull. Amer. Math. Soc. 79(1973), 52–56.Google Scholar
3. Daverman, R.J., Homotopy classification of complements of locally flat codimension two spheres, Amer. J. Math. 98(1976), 367–374.Google Scholar
4. Ferry, S.C., Homotoping t-maps to homeomorphisms, Amer. J. Math. 101(1979), 567–582.Google Scholar
5. Freedman, M., The disk theorem for four-dimensional manifolds. In: Proceedings of the International Congress of Mathematicians, Polish Scientific Publishers, Warsaw, 1984, 647–663.Google Scholar
6. Freedman, M.H. and Quinn, F., Topology of 4-manifolds, Princeton University Press, Princeton 1990.Google Scholar
7. Guilbault, C., An open collar theorem for 4-manifolds, Trans. Amer. Math. Soc. 331(1992), 227–245.Google Scholar
8. Hirschhorn, P.S. and Ratcliffe, J.G., A simple proof of the algebraic unknotting of spheres in codimension two, Amer. J. Math. 102(1980), 489–491.Google Scholar
9. Hollingsworth, J.G. and Rushing, T.B., Homotopy characterization of weakly flat codimension 2 spheres, Amer. J. Math. 98(1976), 385–394.Google Scholar
10. Levine, J., Unknotting spheres in codimension two, Topology 4(1965), 9–16.Google Scholar
11. Liem, V.T. and Venema, G.A., Characterization of knot complements in the 4-sphere, Topology and its Appl. 42(1991), 231–245.Google Scholar
12. Liem, V.T. and Venema, G.A., Complements of 2-spheres in 4-manifolds. In: Topology, Hawaii, to appear.Google Scholar
13. Milnor, J., Infinite cyclic coverings. In: Conference on the Topology of Manifolds, Prindle, Weber and Schmidt, Boston, 1968,115-133.Google Scholar
14. Papakyriakopoulos, C.D., On Dehn's lemma and the asphericity of knots, Ann. Math. 66(1957), 1–26.Google Scholar
15. Steenrod, N.E., Homology with local coefficients, Ann. of Math. 44(1943), 610–627.Google Scholar
16. Sumners, D.W., Homotopy torsion in codimension two knots, Proc. Amer. Math. Soc. 24(1970), 229–240.Google Scholar
17. Sumners, D.W., On asphericity of knots of S2 in S4, unpublished manuscript, 1974.Google Scholar
18. C, C.T.. Wall, Finiteness conditions for CW complexes, Ann. of Math. 81(1965), 56–69.Google Scholar
19. West, J.E., Mapping Hilbert cube manifolds to ANR's: a solution of a conjecture ofBorsuk, Ann. of Math. 106(1977), 1–18.Google Scholar