Pachners

Pachners, in mathematics, refers to the Pachner moves, a collection of local transformations used to modify triangulations of piecewise-linear (PL) manifolds. They are also known as bistellar flips. The moves are named after the Austrian mathematician Udo Pachner, who introduced them in the 1980s, and they provide a canonical set of operations to relate different triangulations of the same PL manifold.

In any fixed dimension d, there are exactly d+1 types of Pachner moves. Each move replaces a triangulation of a d-ball that consists of k d-simplices by a triangulation of the same ball that consists of d+2−k d-simplices, with k ranging from 1 to d+1. The two triangulations agree on the common boundary. Examples: in two dimensions, the moves are 1-3, 3-1, and 2-2; in three dimensions, the moves are 1-4, 4-1, 2-3, and 3-2.

A central result is Pachner’s theorem: any two triangulations of a closed (compact without boundary) PL manifold are connected by a finite sequence of Pachner moves. Variants of the theorem extend to manifolds with boundary under appropriate conditions. The moves are local, do not change the underlying PL homeomorphism type, and thus serve as a powerful tool in PL topology.

Applications of Pachner moves appear in topology, geometry, computational geometry, and theoretical physics. They underpin invariance proofs for PL-manifold invariants, assist in mesh generation and refinement, and appear in state-sum models of quantum gravity where triangulation independence is desired.