Pachner

Pachner refers to Helmut Pachner, an Austrian mathematician renowned for introducing a universal set of local moves on triangulations in piecewise-linear topology. The moves, now called Pachner moves, provide a way to modify a triangulation without changing the underlying PL manifold.

In dimension d, there are d+1 distinct Pachner moves, often described as bistellar flips. Each move replaces a cluster of simplices that triangulates the boundary of a (d+1)-simplex with the triangulation of the boundary of the complementary (d+1)-simplex, and vice versa. The operations are local, finite, and reversible, and they act on triangulations of the same manifold. In three dimensions, the moves include the 1-4 and 2-3 moves and their inverses, 4-1 and 3-2 moves, illustrating the general pattern.

Pachner's theorem states that any two triangulations of a compact PL manifold can be connected by a finite sequence of Pachner moves. This result provides a constructive bridge between different triangulations and underpins many theoretical and algorithmic approaches in topology.

Applications of Pachner moves span computational topology, where they enable algorithms to relate and transform triangulations, to theoretical physics, notably in Regge calculus and spin-foam models, where different triangulations represent discretizations of spacetime. The network formed by triangulations connected through Pachner moves is sometimes studied via a Pachner graph, highlighting the connectivity and transformation pathways between triangulations.