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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The fundamental ideal and $ \pi \sb{2}$ of higher dimensional knots


Author: S. J. Lomonaco
Journal: Proc. Amer. Math. Soc. 38 (1973), 431-433
MSC: Primary 57C45
MathSciNet review: 0339193
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Abstract: Let $ ({S^4},k({S^2}))$ be a knot formed by spinning a polyhedral arc $ \alpha $ about the standard $ 2$-sphere $ {S^2}$ in the $ 3$-sphere $ {S^3}$. Then the second homotopy group of $ {S^4} - k({S^2})$ as a $ Z{\pi _1}$-module is isomorphic to each of the following:

(1) The fundamental ideal modulo the left ideal generated by $ a - 1$, where $ a$ is the image in $ {\pi _1}({S^4} - k({S^2}))$ of a generator of $ {\pi _1}({S^2} - \alpha )$.

(2) The first homology group of the kernel of $ {\pi _1}({S^3} - k({S^2})) \to {\pi _1}({S^4} - k({S^2}))$


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DOI: https://doi.org/10.1090/S0002-9939-1973-0339193-7
Keywords: Spun knot, homotopy groups, homology groups, integral group ring, fundamental ideal, universal covering, fibration, Mayer-Vietoris sequence, Eilenberg-Mac Lane space
Article copyright: © Copyright 1973 American Mathematical Society