# Pachner moves

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 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.