Pachner moves - Alchetron, The Free Social Encyclopedia

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 Δ n + 1 be the ( n + 1 ) -simplex. ∂ Δ 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 ⊂ N together with a simplicial isomorphism ϕ : C → C ′ ⊂ ∂ Δ n + 1 , the Pachner move on N associated to C is the triangulated manifold ( N ∖ C ) ∪ ϕ ( ∂ Δ n + 1 ∖ C ′ ) . By design, this manifold is PL-isomorphic to N but the isomorphism does not preserve the triangulation.

References

Pachner moves Wikipedia

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