On $2$-knot groups with abelian commutator subgroups
HTML articles powered by AMS MathViewer
- by Katsuyuki Yoshikawa
- Proc. Amer. Math. Soc. 92 (1984), 305-310
- DOI: https://doi.org/10.1090/S0002-9939-1984-0754727-3
- PDF | Request permission
Abstract:
In this paper it is shown that if the commutator subgroup of a $2$-knot group is abelian, then it is isomorphic to $Z \oplus Z \oplus Z$, ${Z_\alpha }$, $Z[1/2]$ or $Z[1/2] \oplus {Z_5}$, where $\alpha$ is an odd integer and $Z[1/2]$ is the additive group of the dyadic rationals.References
- Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1985 (French). Algèbre commutative. Chapitres 1 à 4. [Commutative algebra. Chapters 1–4]; Reprint. MR 782296
- R. H. Crowell, The group $G’/G''$ of a knot group $G$, Duke Math. J. 30 (1963), 349–354. MR 154277
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- M. A. Gutiérrez, An exact sequence calculation for the second homotopy of a knot, Proc. Amer. Math. Soc. 32 (1972), 571–577. MR 322848, DOI 10.1090/S0002-9939-1972-0322848-7
- J. C. Hausmann and M. Kervaire, Sous-groupes derivés des groupes de noeuds, Enseign. Math. (2) 24 (1978), no. 1-2, 111–123 (French). MR 488072
- Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 32641, DOI 10.1112/jlms/s1-24.4.247
- Heinz Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), 257–309 (German). MR 6510, DOI 10.1007/BF02565622
- Taizo Kanenobu, Nonribbon $n$-knots with Seifert manifolds homeomorphic to punctured $S^{n}\times S^{1}$, Math. Sem. Notes Kobe Univ. 10 (1982), no. 1, 69–74. MR 672938
- Michel A. Kervaire, Les nœuds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225–271 (French). MR 189052
- Jerome Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1–50. MR 461518, DOI 10.1090/S0002-9947-1977-0461518-0
- J. Levine, Some results on higher dimensional knot groups, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 243–273. With an appendix by Claude Weber. MR 521737
- 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 W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Wiley, New York, 1966.
- D. G. Northcott, A first course of homological algebra, Cambridge University Press, London, 1973. MR 0323867
- Elvira Strasser Rapaport, On the commutator subgroup of a knot group, Ann. of Math. (2) 71 (1960), 157–162. MR 116047, DOI 10.2307/1969883
- Derek John Scott Robinson, A course in the theory of groups, Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York-Berlin, 1982. MR 648604
Bibliographic Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 305-310
- MSC: Primary 57Q45
- DOI: https://doi.org/10.1090/S0002-9939-1984-0754727-3
- MathSciNet review: 754727