Pachner moves

In topology, a branch of mathematics, Pachner moves, named after Udo Pachner, are ways of replacing a triangulation of a piecewise linear manifold by a different triangulation of a homeomorphic manifold. Pachner moves are also called bistellar flips. Any two triangulations of a piecewise linear manifold are related by a finite sequence of Pachner moves.

Let ${\displaystyle \Delta _{n+1}}$ be the ${\displaystyle (n+1)}$-simplex. ${\displaystyle \partial \Delta _{n+1}}$ is a combinatorial n-sphere with its triangulation as the boundary of the n+1-simplex.

Given a triangulated piecewise linear (PL) n-manifold ${\displaystyle N}$, and a co-dimension 0 subcomplex ${\displaystyle C\subset N}$ together with a simplicial isomorphism ${\displaystyle \phi :C\to C'\subset \partial \Delta _{n+1}}$, the Pachner move on N associated to C is the triangulated manifold ${\displaystyle (N\setminus C)\cup _{\phi }(\partial \Delta _{n+1}\setminus C')}$. By design, this manifold is PL-isomorphic to ${\displaystyle N}$ but the isomorphism does not preserve the triangulation.