Kirby Problem 4.53

Definitions and mathematical statement

A finite triangulation is encoded by finitely many vertices and simplices, together with their face incidences. Its geometric realization $|K|$ is obtained by gluing standard Euclidean simplices along their specified faces. The inputs here are promised to be triangulations of closed connected $4$-manifolds; the task does not include deciding whether an arbitrary finite complex is a manifold.

Does there exist a single algorithm which, on every such input $K$, halts after finitely many steps and answers yes exactly when $|K|$ is homeomorphic to the standard topological sphere $S^4$? A homeomorphism is a continuous bijection with continuous inverse. No homeomorphism needs to be supplied with the input, and no bound on the number of simplices is fixed in advance.

This is recognition in the topological category. The question asks for homeomorphism to $S^4$, rather than diffeomorphism to a prescribed smooth sphere. The input is not promised to be simply connected, to have the homology of a sphere, or to lie in any special construction family.

A procedure which eventually verifies all positive instances but may run forever on negative ones is insufficient. Likewise, a collection of computable obstructions which fails to settle some inputs does not decide recognition. The running time may be arbitrarily large, but termination and correctness are required for every valid finite triangulation.

Short English statement

Can one terminating algorithm recognize the topological four-sphere among all closed triangulated four-manifolds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2929 (KP-4.53), 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 4.53. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.