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Transactions of the American Mathematical Society

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Cyclic actions on lens spaces


Author: Paik Kee Kim
Journal: Trans. Amer. Math. Soc. 237 (1978), 121-144
MSC: Primary 57S25
DOI: https://doi.org/10.1090/S0002-9947-1978-0479366-5
MathSciNet review: 479366
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Abstract: A 3-dimensional lens space $ L = L(p,q)$ is called symmetric if $ {q^2} \equiv \pm 1 \bmod p$. Let h be an orientation-preserving PL homeomorphism of even period $ n( > 2)$ on L with nonempty fixed-point set. We show:

(1) If n and p are relatively prime, up to weak equivalence (PL), there exists exactly one such h if L is symmetric, and there exist exactly two such h if L is nonsymmetric.

(2) $ {\text{Fix}}(h)$ is disconnected only if $ p \equiv 0 \bmod n$, and there exists exactly one such h up to weak equivalence (PL).

A $ {Z_n}$-action is called nonfree if $ {\text{Fix}}(\phi ) \ne \emptyset $ for some $ \phi ( \ne 1) \in {Z_n}$. We also classify all orientation-preserving nonfree $ {Z_4}$-actions (PL) on all lens spaces $ L(p,q)$. It follows that each of $ {S^3}$ and $ {P^3}$ admits exactly three orientation-preserving $ {Z_4}$-actions (PL), up to conjugation.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0479366-5
Keywords: Lens spaces, cyclic group action, periodic homeomorphism, fixed-point set
Article copyright: © Copyright 1978 American Mathematical Society

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