## Cyclic actions on lens spaces

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- by Paik Kee Kim PDF
- Trans. Amer. Math. Soc.
**237**(1978), 121-144 Request permission

## 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

- © Copyright 1978 American Mathematical Society
- 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