Classification of homotopy torus knot spaces
HTML articles powered by AMS MathViewer
- by Richard S. Stevens
- Proc. Amer. Math. Soc. 52 (1975), 461-464
- DOI: https://doi.org/10.1090/S0002-9939-1975-0380807-5
- PDF | Request permission
Abstract:
The existence of nontrivial homotopy torus knot spaces is established as a corollary to the Theorem. Let $p$ and $q$ be two integers with $p > 1,q > 1$, and $(p,q) = 1$. Let $\mathfrak {M}$ be a maximal set of topologically distinct compact orientable irreducible $3$-mainfolds with fundamental group presented by $\langle a,b|{a^p}{b^q}\rangle$. Then card $(\mathfrak {M}) = 1/2\Phi (pq)$, where $\Phi$ denotes Euler’s function.References
- H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, New York-Heidelberg, 1972. MR 0349820
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
- R. P. Osborne and R. S. Stevens, Group presentations corresponding to spines of $3$-manifolds. I, Amer. J. Math. 96 (1974), 454–471. MR 356058, DOI 10.2307/2373554
- Richard S. Stevens, Classification of $3$-manifolds with certain spines, Trans. Amer. Math. Soc. 205 (1975), 151–166. MR 358786, DOI 10.1090/S0002-9947-1975-0358786-0
- Friedhelm Waldhausen, Gruppen mit Zentrum und $3$-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505–517 (German). MR 236930, DOI 10.1016/0040-9383(67)90008-0
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 52 (1975), 461-464
- MSC: Primary 57A35; Secondary 55A05
- DOI: https://doi.org/10.1090/S0002-9939-1975-0380807-5
- MathSciNet review: 0380807