Existence of infinitely many genuine unflippable 3-sphere triangulations

Determine whether infinitely many complexes in the proposed family of unflippable complexes are genuine triangulations of the 3-sphere rather than triangulations of other topological spaces.

Background

The paper discusses an unpublished construction suggesting an infinite family of unflippable complexes, meaning complexes admitting neither 2–3 nor 3–2 bistellar flips. The known examples are isolated vertices in the vertex-preserving flip graphs of triangulated 3-spheres.

The unresolved issue is whether infinitely many members of the suggested family actually have the topology of the 3-sphere. Resolving this would establish whether the observed fragmentation of the flip graphs reflects an infinite phenomenon within 3-sphere triangulations or instead arises partly from complexes of other topological types.

References

Dougherty’s unpublished work further suggests an infinite family of such unflippable complexes --- those admitting neither 2--3 nor 3--2 flips --- though it remains open whether infinitely many of these are genuine triangulated 3-spheres or triangulations of other topologies.

— Components of Flip Graph of Triangulated S^3  (2505.06472 - Faber et al., 10 May 2025) in Section 1, Introduction