Topologically but not smoothly integrally split 2-component links in $S^4$ (Golla–Marengon Questions 1.4 and 1.5)

For 2-component links of 2-spheres in $S^4$, being split can be generalized: a link is split by a closed 3-manifold $Y$ if an embedded copy of $Y$ separates the components, and integrally split if this holds for some integer homology 3-sphere. Golla and Marengon (2024; IMRN 2025) constructed links integrally split but not split by $S^3$, with Heegaard Floer obstructions, but left the categorical gap open: they ask (Question 1.4) whether a 2-component link in $S^4$ exists that is topologically but not smoothly integrally split — separated by a locally flat integer homology 3-sphere but by no smooth one — and (Question 1.5) the relative version for a given integer homology sphere $Y$. Nersisyan (2026) realizes the analogous topological-but-not-smooth splitting only for codimension-zero links of Mazur manifolds in $\#^{n}\mathbb{CP}^{2}$, $n\neq 0$, and reports that the $S^4$ case remains unachieved. Both questions are open.

Background

A surface link or 2-link in $S^4$ is the image of an embedding of a disjoint union of 2-spheres, considered up to ambient isotopy; a 2-component link is split if some embedded 3-sphere separates the two components into different regions of the complement. Both notions have a smooth version (smooth embeddings) and a topological version (locally flat embeddings, i.e. embeddings that look standard near every point). As in the source, the link itself is taken to be smoothly embedded, and the smooth/topological labels attach to the embedded separating manifold. Golla and Marengon generalized the notion as follows: a link in $S^{n+1}$ is split by a closed $n$-manifold $Y$ if some (smooth or locally flat) embedded copy of $Y$ in $S^{n+1}$ separates the components, and a link is integrally split if it is split by some integer homology $n$-sphere, a closed $n$-manifold with the same integer homology groups as $S^n$ (for $n=3$: an integer homology 3-sphere, or $\mathbb{Z}HS^3$). The question originates with Peter Kronheimer, posed after a talk of Maggie Miller on non-isotopic splitting spheres for split 2-links (Hughes–Kim–Miller), and asks how far integral splitting differs from ordinary splitting and how the smooth and topological categories diverge.

Golla and Marengon proved three things. (Theorem 1.1) For every $n\geq 3$ there is a 2-component link in $S^{n+1}$ that is smoothly integrally split but not even topologically split by $S^n$, so integral splitting is strictly weaker than ordinary splitting. (Theorem 1.2) In dimension four there are infinite families of links $L_k$ and integer homology 3-spheres $Y_k$ such that $L_k$ is smoothly split by $Y_k$ but not topologically split by any other $Y_\ell$. (Theorem 1.3) A Heegaard Floer obstruction: Heegaard Floer homology is an invariant of Ozsváth and Szabó assigning graded modules such as $\mathrm{HF}^+$ to closed 3-manifolds, and $\mathrm{HF}_{\mathrm{red}}$ is its reduced part, which measures how far the manifold is from the simplest ones (it vanishes for lens spaces); if a link $L\subset S^4$ is smoothly split by a $\mathbb{Z}HS^3$ $Y$ and the double branched cover $\Sigma_2(S^4,L)$ (the two-fold cover of $S^4$ branched along the link, a standard invariant of surface links) is a mapping torus of a diffeomorphism of $Y$ (the quotient of $Y\times[0,1]$ obtained by regluing the two ends via the diffeomorphism), then every other $\mathbb{Z}HS^3$ $Z$ that smoothly splits $L$ must contain $\mathrm{HF}_{\mathrm{red}}(Y)$ as a direct summand of $\mathrm{HF}_{\mathrm{red}}(Z)$.

All of these results concern links that are smoothly split by some homology sphere, with obstructions to splitting by other specific homology spheres; none produces a link for which the topological and smooth notions of integral splitting differ, and the authors pose exactly this as their closing questions. They also note (Remark 3.2) that no strengthening to nonstandard homotopy spheres is possible for $n\neq 4$, since nonstandard homotopy $n$-spheres do not embed in $S^{n+1}$ (Smale), while in dimension four the possibility of splitting by an exotic $S^4$, if one exists, is entangled with the still-open smooth 4-dimensional Poincaré conjecture. The problem below is the explicitly stated open target of that paper. In the relative question (1.5) the integer homology sphere $Y$ is fixed in advance and is understood to admit a smooth embedding in $S^4$: otherwise “not smoothly split by $Y$” would hold automatically, and the question would degenerate into a statement about which homology spheres embed smoothly in $S^4$ rather than about the link.

Problem Statement

Does there exist a 2-component link of 2-spheres in $S^4$ that is topologically but not smoothly integrally split — that is, a link for which some locally flat embedded integer homology 3-sphere separates the two components, but no smoothly embedded integer homology 3-sphere does (Golla–Marengon, Question 1.4)?

Together with its relative companion (Golla–Marengon, Question 1.5). The source asks: “Given an integer homology sphere $Y$, does there exist a 2-component link in $S^4$ that is topologically but not smoothly split by $Y$?” Here $Y$ is fixed in advance as a parameter of the question, not a manifold selected as part of an answer: does there exist a 2-component link in $S^4$ that is topologically split by the given $Y$ (some locally flat embedded copy of $Y$ separates the components) but not smoothly split by $Y$ (no smooth embedded copy of $Y$ separates them)?

The two questions are distinct: Question 1.4 ranges over all integer homology 3-spheres as potential smooth splittings, while in Question 1.5 the splitting manifold $Y$ is given in advance. Neither question implies the other, and both are in scope.

A resolution of these questions is either an explicit link realizing the gap — with a proof of the topological splitting and a smooth obstruction valid against all smooth integer homology 3-spheres, or, in the relative version, against the given $Y$ — or, conversely, a rigidity proof that topological integral splitting implies smooth integral splitting for 2-component links in $S^4$ (respectively, that topological splitting by the given $Y$ implies smooth splitting by $Y$).

References

1. Marco Golla and Marco Marengon, Splitting Links by Integer Homology Spheres, International Mathematics Research Notices (2025), DOI: 10.1093/imrn/rnaf309; preprint arXiv:2403.00064, https://arxiv.org/abs/2403.00064
2. Sergey Nersisyan, Links of Mazur manifolds and exotica, preprint arXiv:2606.24220 (2026), DOI: 10.48550/arXiv.2606.24220, https://arxiv.org/abs/2606.24220
3. Mark Hughes, Seungwon Kim, Maggie Miller, Non-isotopic splitting spheres for a split link in $S^{4}$, Proceedings of the London Mathematical Society (2025), DOI: 10.1112/plms.70038; preprint arXiv:2307.12140, https://arxiv.org/abs/2307.12140