Pachner moves

File:Pachner Move.png

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.

References

{{citation|first=Udo|last=Pachner|title=P.L. homeomorphic manifolds are equivalent by elementary shellings|journal=European Journal of Combinatorics|volume=12|issue=2|year=1991|pages=129–145|doi=10.1016/s0195-6698(13)80080-7|doi-access=free}}.

Category:Topology

Category:Geometric topology

Category:Structures on manifolds

Pachner moves

Definition

See also

References