Pachner moves: Difference between revisions

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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 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

Pachner, Udo (1991), "P.L. homeomorphic manifolds are equivalent by elementary shellings", European Journal of Combinatorics, 12 (2): 129–145, doi:10.1016/s0195-6698(13)80080-7.

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 Let <math>\Delta_{n+1}</math> be the <math>(n+1)</math>-[[simplex]].   <math>\partial \Delta_{n+1}</math> is a combinatorial ''n''-sphere with its triangulation as the boundary of the ''n+1''-simplex.
-Given a triangulated piecewise linear ''n''-manifold <math>N</math>, and a co-dimension ''0'' subcomplex <math>C \subset N</math> together with a simplicial isomorphism <math>\phi : C \to C' \subset \partial \Delta_{n+1}</math>, the Pachner move on ''N'' associated to ''C'' is the triangulated manifold <math>(N \setminus C) \cup_\phi (\partial \Delta_{n+1} \setminus C')</math>. By design, this manifold is PL-isomorphic to <math>N</math> but the isomorphism does not preserve the triangulation.
+Given a triangulated piecewise linear (PL) ''n''-manifold <math>N</math>, and a co-dimension ''0'' subcomplex <math>C \subset N</math> together with a simplicial isomorphism <math>\phi : C \to C' \subset \partial \Delta_{n+1}</math>, the Pachner move on ''N'' associated to ''C'' is the triangulated manifold <math>(N \setminus C) \cup_\phi (\partial \Delta_{n+1} \setminus C')</math>. By design, this manifold is PL-isomorphic to <math>N</math> but the isomorphism does not preserve the triangulation.
 ==See also==