The census study of Budney and Burton on smooth embeddings of closed 3-manifolds in the 4-sphere posed a two-part question comparing embedding targets: if a 3-manifold admits a smooth embedding into a homotopy 4-sphere, must it admit one into the standard 4-sphere; and do there exist 3-manifolds that embed in homology 4-spheres but not in the 4-sphere. The second part is now resolved - McDonald constructed infinitely many integer homology 3-spheres that embed in integer homology 4-spheres but in no homotopy 4-sphere. This record states the surviving open core: must every closed 3-manifold that embeds smoothly in some homotopy 4-sphere also embed smoothly in the standard S4S^4? Every known embedding obstruction vanishes identically on all integral homology 4-spheres, and four census manifolds, including the Brieskorn sphere Σ(2,3,25)\Sigma(2,3,25), embed in homotopy 4-spheres with no known standard-sphere embedding. A negative answer would exhibit an exotic smooth 4-sphere and refute the smooth 4-dimensional Poincare conjecture; a positive answer would be a new structural theorem on the transfer of smooth embeddings across smooth structures.

Background

A closed orientable 3-manifold always admits a smooth embedding in the 5-sphere, so for such a manifold MM other than S3S^3 the smooth embedding dimension is either 4 or 5, and the borderline case is genuinely hard: deciding whether a given MM embeds smoothly in S4S^4 is a long-standing open problem in low-dimensional topology. If MM does embed, it separates the sphere into two compact 4-manifolds S4=V1∪MV2S^4 = V_1 \cup_M V_2 meeting along their common boundary MM, and Poincare-Lefschetz duality then forces strong algebraic conditions on MM: the torsion subgroup of H1(M;Z)H_1(M;\mathbb{Z}), equipped with its classical linking form (a nonsingular pairing valued in Q/Z\mathbb{Q}/\mathbb{Z} defined by linking torsion cycles), must be hyperbolic, meaning it splits as an orthogonal direct sum of two self-annihilating summands, each summand coinciding with the orthogonal complement of the other. This torsion linking-form test already rules out, for example, every nontrivial lens space (a quotient of S3S^3 by a free linear action of a cyclic group). An integral homology 4-sphere is a closed 4-manifold with the same integral homology groups as S4S^4; a homotopy 4-sphere is a closed smooth 4-manifold homotopy equivalent to S4S^4. The resolution of the topological 4-dimensional Poincare conjecture (1982) shows that every homotopy 4-sphere is homeomorphic to S4S^4, but whether every one is diffeomorphic to S4S^4 is the smooth 4-dimensional Poincare conjecture, which remains open; a counterexample is called an exotic 4-sphere. Crucially, the classical algebraic obstructions such as the linking-form test obstruct embedding into every integral homology 4-sphere, not into S4S^4 specifically, so they are blind to the difference between these targets. Embeddings are constructed by Dehn surgery on links in S3S^3 - removing a tubular neighborhood of each link component and regluing the solid torus so that the curve prescribed by an integer surgery coefficient bounds a disc. A slice link is a link in S3S^3 whose components bound disjoint smoothly embedded 2-discs in the 4-ball. Surgery with coefficient 0 on a link that splits into two disjoint slice sublinks yields a 3-manifold that embeds in S4S^4; surgery with coefficients ±1\pm 1 on a slice link yields a smooth embedding into a homotopy 4-sphere, because the surgery trace is a contractible 4-manifold with boundary MM and the double of a contractible 4-manifold is a homotopy 4-sphere containing MM as its equator. Similarly, Brieskorn spheres - the integer homology 3-spheres Σ(p,q,r)\Sigma(p,q,r) arising as links of the complex surface singularity xp+yq+zr=0x^p + y^q + z^r = 0 - embed in homotopy 4-spheres whenever they bound smooth contractible 4-manifolds, as happens in several classical families. For an integer homology 3-sphere YY, bounding a smooth contractible 4-manifold is a sufficient but not a necessary condition for embedding smoothly in some homotopy 4-sphere. Sufficiency holds because the double of a contractible filling is a homotopy 4-sphere containing YY as its equator; necessity fails because the complementary pieces of a smooth embedding of a homology 3-sphere in a homotopy 4-sphere are acyclic by Mayer-Vietoris and duality - each has the integral homology of a point - and acyclic does not imply contractible in dimension 4. The contractible-filling condition is therefore strictly stronger than embedding in a homotopy 4-sphere. Budney and Burton tested this landscape computationally on a published census of the 13766 triangulated prime closed 3-manifolds built from at most 11 tetrahedra (13400 of them orientable). Only 149 of the orientable manifolds have hyperbolic torsion linking forms and hence pass the basic test for embedding in S4S^4: 41 were known to embed in S4S^4, four more were given embeddings into homotopy 4-spheres with no known embedding into S4S^4 (among them Σ(2,3,25)\Sigma(2,3,25), which bounds a smooth contractible 4-manifold), 67 were obstructed from embedding in any homology 4-sphere, and 37 cases were left unresolved. Struck by the observation that all their obstructions applied equally to every homology 4-sphere - and that no 3-manifold was known to embed in a homology 4-sphere yet fail to embed in S4S^4 - they posed the two-part Question 8.3 recorded here.

Problem Statement

Question 8.3 of Budney and Burton poses two linked questions. The second - whether there exist 3-manifolds that embed in homology 4-spheres but not in S4S^4 - has been resolved affirmatively by McDonald, so the open problem is the first, stated in its original generality: If a closed 3-manifold MM admits a smooth embedding into some homotopy 4-sphere, must MM admit a smooth embedding into the standard S4S^4 (equivalently, into R4\mathbb{R}^4)? For integer homology 3-spheres this yields a restricted sub-case rather than an equivalent formulation: since bounding a smooth contractible 4-manifold implies embedding in some homotopy 4-sphere but not conversely, one may ask whether a homology 3-sphere that bounds a smooth contractible 4-manifold must embed smoothly in S4S^4. Resolving this sub-case would not by itself settle the general question, which quantifies over all closed 3-manifolds that embed in some homotopy 4-sphere. The logical stakes give the problem a sharp boundary: a negative answer produces a 3-manifold that embeds only into nonstandard homotopy 4-spheres, thereby exhibiting an exotic smooth 4-sphere and refuting the smooth 4-dimensional Poincare conjecture; a positive answer is implied by, but does not imply, that conjecture. A resolution must hold for arbitrary closed 3-manifolds; positive results restricted to particular families of homotopy 4-spheres or of 3-manifolds remain partial.

References

  1. Ryan Budney and Benjamin A. Burton, Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census, arXiv:0810.2346 (v6, 2020); published as Experimental Mathematics 31 (2022), no. 3, 988-1013, DOI 10.1080/10586458.2020.1740836, https://arxiv.org/abs/0810.2346
  2. Clayton McDonald, Surface slices and homology spheres, Geometry & Topology 30 (2026), no. 5, 1611-1623, DOI 10.2140/gt.2026.30.1611, arXiv:2202.02696, https://arxiv.org/abs/2202.02696
  3. Ian Agol and Mike Freedman, Embedding Heegaard Decompositions, New Zealand Journal of Mathematics 52 (2021), 727-731, DOI 10.53733/189, arXiv:1906.03244, https://arxiv.org/abs/1906.03244
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  6. Jae Choon Cha and Min Hoon Kim, Calegari's homotopy 4-spheres from fibered knots are standard, arXiv:2411.10051 (2024), https://arxiv.org/abs/2411.10051
  7. Clayton McDonald, An S^2 x S^2 embedding, arXiv:2607.27619 (2026), https://arxiv.org/abs/2607.27619
  8. Paolo Aceto, Marco Golla, and Kyle Larson, Embedding 3-manifolds in spin 4-manifolds, Journal of Topology 10 (2017), 301-323, DOI 10.1112/topo.12010, arXiv:1607.06388, https://arxiv.org/abs/1607.06388
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