Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T22:48:29.040Z Has data issue: false hasContentIssue false
  • English
  • Français

High dimensional knot groups which are not two-knot groups

Published online by Cambridge University Press:  17 April 2009

Jonathan A. Hillman
Jonathan A. Hillman
Affiliation:
Department of Mathematics, School of General Studies, Australian National University, Canberra, ACT.
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.

This paper presents three arguments, one involving orientability, and the others Milnor duality and, respectively, the injectivity of cup product into H2 for an abelian group and free finite group actions on homotopy 3-spheres to show that there are high dimensional knot groups which are not the groups of knotted 2-spheres in S4, thus answering a question of Fox (“Some problems in knot theory”, Topology of 3-manifolds and related topics”, 168–176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 16 , Issue 3 , June 1977 , pp. 449 - 462
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Cappell, Sylvian E., “Superspinning and knot complements”, Topology of manifolds, 358–383 (Proc. Univ. of Georgia Topology of Manifolds Inst. 1969. Markham, Chicago, 1970).Google Scholar
[2]Cappell, Sylvian E. and Shaneson, Julius L., “Some new four-manifolds”, Ann. of Math. (2) 104 (1976), 61–72.Google Scholar
[3]Cartan, Henri and Eilenberg, Samuel, Homological algebra (Princeton Mathematical Series, 19. Princeton University Press, Princeton, New Jersey, 1956).Google Scholar
[4]Cohen, M.M., A course in simple-homotopy theory (Graduate Texts in Mathematics, 10. Springer-Verlag, New York, Heidelberg, Berlin, 1973).Google Scholar
[5]Fox, R.H., “A quick trip through knot theory”, Topology of 3-manifolds and related topics, 120–167 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).Google Scholar
[6]Fox, R.H., “Some problems in knot theory”, Topology of 3-manifolds and related topics, 168–176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).Google Scholar
[7]Giffen, Charles H., “On aspherical embeddings of 2-spheres in the 4-sphere”, Topology Seminar, Wisconsin, 1965, 189–165 (Annals of Mathematics Studies, 60. Princeton University Press, Princeton, New Jersey, 1966).Google Scholar
[8]Hu, Sze-Tsen, Homotopy theory (Pure and Applied Mathematics, 8. Academic Press, New York and London, 1959).Google Scholar
[9]Huppert, B., Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).Google Scholar
[10]Kervaire, Michel A., “Les nœuds de dimensions supérieures”, Bull. Soc Math. France 93 (1965), 225–271.Google Scholar
[11]Mazur, Barry, “Symmetric homology spheres”, Illinois J. Math. 6 (1962), 245–250.CrossRefGoogle Scholar
[12]Milnor, John, “Groups which act on Sn without fixed points”, Amer. J. Math. 79 (1957), 623–630.CrossRefGoogle Scholar
[13]Milnor, John W., “Infinite cyclic coverings”, Conference on the topology of manifolds, 115–133 (The Prindle, Weber & Schmidt Complementary Series in Mathematics, 13. Prindle, Weber & Schmidt, Boston, Massachusetts; London; Sydney; 1968).Google Scholar
[14]Newman, Morris, Integral matrices (Pure and Applied Mathematics, 45. Academic Press, New York and London, 1972).Google Scholar
[15]Papakyriakopoulo, C.D., “On Dehn's lemma and the asphericity of knots”, Ann. of Math. (2) 66 (1957), 1–26.Google Scholar
[16]Shaneson, Julius L., “Embeddings with codimension two of spheres in spheres and H-cobordisms of S 1 × S 3”, Bull. Amer. Math. Soc. 74 (1968), 972–974.Google Scholar
[17]Stallings, John, Group theory and three-dimensional manifolds (Yale Mathematical Monographs, 4. Yale University Press, New Haven and London, 1971).Google Scholar
[18]Sullivan, Dennis, “On the intersection ring of compact three manifolds”, Topology 14 (1975), 275–277.CrossRefGoogle Scholar
[19]Zeeman, E.G., “Twisting spun knots”, Trans. Amer. Math. Soc. 115 (1965), 471–495.Google Scholar