Pachner moves

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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 homoeomorphic 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[edit]

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.

References[edit]

  • Pachner, Udo (1991), "P.L. homeomorphic manifolds are equivalent by elementary shellings", European Journal of Combinatorics 12 (2): 129–145 .