Recognizability of the four-sphere

Determine whether the four-sphere is recognizable among closed four-manifolds; that is, determine whether an algorithm can decide from a triangulation of a closed four-manifold whether it is homeomorphic to $S^4$.

Background

A closed manifold is recognizable if an algorithm can decide, from a triangulation of a closed manifold, whether it is homeomorphic to the specified manifold. The paper contrasts the established recognizability results in dimensions at most three with the existence of unrecognizable manifolds in dimensions at least four.

The four-dimensional case is described as substantially less understood. Although the paper proves topological and smooth unrecognizability for connected sums of sufficiently many copies of S2×S2S^2\times S^2, it does not resolve the corresponding question for the four-sphere itself.

References

It is a major open question whether $S4$ is recognizable; see, for example, Problem~4.53 in the K3 problem list .

— Small undecidable groups and unrecognizable 4-manifolds  (2609.10461 - Kegel et al., 9 Sep 2026) in Section 1, subsection “Topology”