Uniqueness of free 2-periodicities of links

TL;DR

The paper proves that for links in $\mathbb{R}\mathbb{P}^{3}$, isotopy of their lifts under the double cover $\pi: S^{3} \to \mathbb{R}\mathbb{P}^{3}$ implies isotopy of the original links, establishing the uniqueness of free 2-periodicity of links in $S^{3}$. The argument proceeds by a case analysis of the link complements: hyperbolic, Seifert-fibered, and then general via prime and JSJ decompositions, leveraging Mostow rigidity, the spherical space form conjecture, and JSJ-structure arguments. A key idea is that outermost JSJ pieces lift to connected pieces in $S^{3}$, allowing an equivariant reembedding and isotopy extension that reduces the problem to the corresponding results in the lifted setting and the solid-torus case treated in prior work. The results unify and extend known rigidity phenomena for free periodicities, clarifying when higher-periodicities may fail to be unique and when they are constrained by the ambient covering structure. Overall, the work provides a robust framework to deduce isotopy of $\mathbb{R}\mathbb{P}^{3}$-links from lifted data in $S^{3}$.

Abstract

We show that if two links in the real projective 3-space $\mathbb{RP}^{3}$ have isotopic preimages in the 3-sphere $S^{3}$ by the double covering map, then they are themselves isotopic in $\mathbb{RP}^{3}$.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 2.1: Spherical space form conjecture
• Lemma 2.2: KMMY25 Lemma 4.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Proposition 3.1: Budney06 Proposition 3.5
• Lemma 3.2