Kirby Problem 4.25

Definitions and mathematical statement

Let $M$ be a closed connected orientable smooth $3$-manifold, and give $S^4$ its standard smooth structure. A smooth embedding $e:M\to S^4$ is a smooth injective immersion which is a homeomorphism onto its image. Since $M$ is compact, the image is a closed embedded hypersurface. No preferred position or parameterization of that image is prescribed.

Find necessary and sufficient conditions on $M$ for such an embedding to exist. The classification should distinguish exactly the manifolds admitting smooth embeddings in the standard $4$-sphere, rather than only supplying necessary obstructions or sufficient constructions for special families.

The algorithmic part asks whether a single algorithm can decide this property from a finite triangulation of $M$. The input is promised to triangulate a closed orientable $3$-manifold. The algorithm must halt on every valid input and return yes precisely when the associated smooth $3$-manifold embeds smoothly in $S^4$. There is no bound specified on its running time.

The target is smooth embeddability; a locally flat topological embedding alone does not suffice. Similarly, embeddability in another homology $4$-sphere, or in a connected sum of copies of $S^2\times S^2$, is a different question. The criterion and decision procedure must work for all closed orientable $M$, without assuming that $M$ is a homology sphere, prime, or given with a special geometric structure.

Short English statement

Characterize the closed orientable three-manifolds that smoothly embed in the standard four-sphere, and decide whether this property is algorithmically decidable.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2901 (KP-4.25), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.25. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.