Smooth $Z_{p}$-actions on spheres which leave knots pointwise fixed

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.