Splitting spheres for unlinked $S^2$'s in $S^4$

TL;DR

The paper proves the existence of infinitely many pairwise non-isotopic splitting spheres for the unlink of two unknotted $S^2$'s in $S^4$ by constructing an infinite family of pairwise non-isotopic $S^3$'s in $S^4 \,\setminus K$, realized via barbell diffeomorphisms in $S^1 × B^3$ and lifted to cyclic covers. The key tool is the Budney–Gabai barbell construction, which yields nontrivial elements in $π_0(Diff_∂)$ after lifting, and a Cerf-type argument that obstructs isotopy to the identity. By transferring these diffeomorphisms into the splitting-sphere setting, the author obtains infinitely many non-isotopic splitting spheres for the unlink, answering a question of Hughes–Kim–Miller. The work connects to broader questions about 4-manifold knotting and smooth Schoenflies phenomena, illustrating that even the simplest unlink case supports rich nontrivial sphere embeddings.

Abstract

We show that there exist infinitely many pairwise non-isotopic splitting spheres for two unlinked, unknotted $S^2$'s in $S^4$. This answers a question posed by Hughes, Kim, and Miller.

Figures (7)

• Figure 1: The barbell representing the diffeomorphism $\delta_k$ of $S^1 \times B^3$, as seen in the $S^1 \times D^2 \times 0$ slice BG.
• Figure 2: $\beta_k$ in the $t=0$ slice of $L\# R$. The numbering on the orange $S^2$'s denotes which points are identified.
• Figure 3: The isotopy of $\hat{\beta_k}$ to $\delta_k.$ Note that we also recover $\delta_k$ if we cap off $L$ rather than $R$.
• Figure 4: Schematic of $f_L \# f_R$.
• Figure 5: The 2-dimensional analogue of $X$ for $k=5$. The central $S^1 \times B^1$ covers $R$, and the orbiting $S^1 \times B^1$'s cover $L$.

Theorems & Definitions (4)

• Definition 1
• Theorem 2
• Remark 3
• proof