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.

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

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