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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The fundamental ideal and $\pi _{2}$ of higher dimensional knots
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by S. J. Lomonaco
Proc. Amer. Math. Soc. 38 (1973), 431-433
DOI: https://doi.org/10.1090/S0002-9939-1973-0339193-7

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 ${\pi _1}({S^3} - k({S^2})) \to {\pi _1}({S^4} - k({S^2}))$
References
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Bibliographic Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 38 (1973), 431-433
  • MSC: Primary 57C45
  • DOI: https://doi.org/10.1090/S0002-9939-1973-0339193-7
  • MathSciNet review: 0339193