Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere?

Definitions and mathematical statement

The input is a finite simplicial complex $K$, presented by its finite vertex and simplex lists, promised to triangulate a closed connected piecewise-linear four-manifold. In particular its vertex links are piecewise-linearly homeomorphic to $S^3$. Let $\partial\Delta^5$ be the boundary complex of the five-simplex, a standard triangulation of the four-sphere.

Is there an algorithm that halts on every promised input and decides whether \[ |K|\cong_{\mathrm{PL}}|\partial\Delta^5|? \] Here PL equivalence means that after suitable finite subdivisions the triangulated spaces admit a piecewise-linear homeomorphism. It does not mean that the original simplex lists are isomorphic. Equivalently one can use existence of a finite sequence of bistellar (Pachner) moves and simplicial isomorphisms joining the triangulations.

The requested procedure has no time-complexity bound, but it must terminate correctly in both cases. Enumerating successful move sequences only recognizes yes-instances and supplies no termination test on no-instances. The input promise avoids conflating recognition of a four-manifold with recognition of the standard PL sphere. The category remains PL throughout: topological homeomorphism or homotopy equivalence alone is not the specified equivalence. The source's discussion of smoothings is unnecessary for stating this exact finite-input decision problem.

Short English statement

Is there an always-terminating algorithm recognizing the standard piecewise-linear four-sphere among finite triangulated closed four-manifolds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3418 (OPG-37161), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere?. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/is_there_an_algorithm_to_determine_if_a_triangulated_4_manifold_is_combinatorially_equivalent_to_the_4_sphere.