Pachner train

This Pachner move replaces two simplices in with the other three. ${\displaystyle \partial \Delta ^{4}}$

The Pachner train is a term from combinatorial topology , i.e. the study of simplicial complexes and triangulated manifolds within mathematics .

A Pachner train replaces some simplices in a triangulated -manifold with other simplices in such a way that the union of the replaced and the replacing simplices exactly forms the edge of a - simplex . ${\displaystyle n}$ ${\displaystyle (n+1)}$

The meaning of the Pachner trains results from the fact that two different triangulations of a manifold can be converted into one another by a sequence of Pachner trains. This was proven by Pachner in 1991.

definition

In the following, denote the -dimensional standard simplex and its edge with the triangulation by side surfaces. ${\displaystyle \Delta ^{n+1}}$ ${\displaystyle (n+1)}$ ${\displaystyle \partial \Delta ^{n+1}}$

A -dimensional triangulated manifold is given . A Pachner move consists in the selection of a sub-complex isomorphic to a -dimensional sub- complex and the formation of the triangulated manifold ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle C^{\prime }\subset \partial \Delta ^{n+1}}$ ${\displaystyle C\subset M}$

${\displaystyle (M\setminus C)\cup _{\phi }(\partial \Delta ^{n+1}\setminus C^{\prime })}$,

where the sticking map is the constraint on the given simplicial isomorphism . ${\displaystyle \phi \colon \partial C\to \partial C^{\prime }}$ ${\displaystyle C\to C^{\prime }}$

By means of this construction one again obtains the same manifold , but with a different triangulation from the original. ${\displaystyle M}$

Examples

In the case of n-dimensional manifolds is called - - and - -Pachner trains. A - -Pachner move replaces one -dimensional simplex with four others (or vice versa), a - -Pachner move replaces two -dimensional simplices with three others (or vice versa). ${\displaystyle 3}$ ${\displaystyle 1}$ ${\displaystyle 4}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 1}$ ${\displaystyle 4}$ ${\displaystyle 3}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 3}$

Pachner's theorem

Pachner's theorem : If two triangulated PL-manifolds (of any dimension) are PL-homeomorphic, then there is a sequence of Pachner trains which converts one triangulation into the other.

In particular, for surfaces and -dimensional manifolds, two triangulations can be converted into one another by a sequence of Pachner trains. (This results from the uniqueness of the PL structure for manifolds of dimensions and .) ${\displaystyle 3}$ ${\displaystyle 2}$ ${\displaystyle 3}$

literature

• Udo Pachner: Homeomorphic manifolds are equivalent by elementary shellings , European J. Combin. 12: 129-145 (1991).
• WBR Lickorish : Simplicial moves on complexes and manifolds. Proceedings of the Kirbyfest (Berkeley, CA, 1998), 299-320 Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999.