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Transactions of the American Mathematical Society

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Smooth $ Z\sb{p}$-actions on spheres which leave knots pointwise fixed


Author: D. W. Sumners
Journal: Trans. Amer. Math. Soc. 205 (1975), 193-203
MSC: Primary 57E25
DOI: https://doi.org/10.1090/S0002-9947-1975-0372893-8
MathSciNet review: 0372893
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Abstract: The paper produces, via handlebody construction, a family of counterexamples to the generalized Smith conjecture; that is, for each pair of integers $ (n,p)$ with $ n \geq 2$ and $ p \geq 2$ there are infinitely many knots $ ({S^{n + 2}},k{S^n})$ which admit smooth semifree $ {Z_p}$-actions (fixed on the knotted submanifold $ k{S^n}$ and free on the complement $ ({S^{n + 2}} - k{S^n}))$. This produces previously unknown $ {Z_p}$-actions on $ ({S^4},k{S^2})$ for $ p$ even, the one case not covered by the work of C. H. Giffen. The construction is such that all of the knots produced are equivariantly null-cobordant. Another result is that if a knot admits $ {Z_p}$ -actions for all $ p$, then the infinite cyclic cover of the knot complement is acyclic, and thus leads to an unknotting theorem for $ {Z_p}$-actions.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0372893-8
Keywords: Generalized Smith conjecture, semifree $ {Z_p}$-actions on knotted sphere and ball pairs, $ {Z_p}$-equivariantly null-cobordant knots, unknotting theorem for $ {Z_p}$-actions, branched cyclic coverings, infinite cyclic coverings, handlebody addition
Article copyright: © Copyright 1975 American Mathematical Society

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