Kirby Problem 3.28

Definitions and mathematical statement

A triangulation of a compact PL three-manifold is a finite simplicial description of that manifold. An interior Pachner move replaces a union of $k$ tetrahedral facets of the boundary of a four-simplex by the complementary $5-k$ facets, for $k=1,2,3,4$, whenever the replacement is admissible. These are the $1$--$4$, $2$--$3$, and inverse moves. For manifolds with boundary also allow the usual elementary boundary shellings and inverse shellings, attaching or removing a tetrahedron along a boundary disk made of its faces.

For triangulations $T_1,T_2$ of the same manifold, let $d_P(T_1,T_2)$ be the least number of allowed moves needed to pass from one to a triangulation combinatorially isomorphic to the other. For a fixed $M$, define \[ P_M(t_1,t_2)=\max_{|T_i|=t_i}d_P(T_1,T_2),\qquad P(t_1,t_2)=\max_M P_M(t_1,t_2), \] where $|T_i|$ counts tetrahedra and maxima range over existing pairs.

How fast do these functions grow? Find effective upper and lower bounds, both universally and for specified manifolds or classes. Every intermediate triangulation may have more tetrahedra than either endpoint. Consequently, bounding only sequences that stay below a fixed tetrahedron count would not bound the unrestricted Pachner distance.

Short English statement

Bound the number of local triangulation moves needed to connect two triangulations of the same compact three-manifold.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2826 (KP-3.28), 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 3.28. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.