Cyclic actions on lens spaces

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.