Simplicial moves on balanced complexes

TL;DR

This paper introduces cross-flips, a new local move for transforming balanced triangulations of manifolds, and proves they can connect any two such triangulations, extending the concept of bistellar flips.

Contribution

The paper defines cross-flips for balanced complexes and proves their sufficiency to connect all balanced triangulations of a closed manifold.

Findings

Cross-flips can transform any balanced triangulation into another.

Any two balanced triangulations of a closed manifold are connected by cross-flips.

Bistellar flips can connect colored triangulations, preserving vertex coloring.

Abstract

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial $d$-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every $m \geq d+2$ and any closed combinatorial $d$-manifold $M$, two $m$-colored triangulations of $M$ can be connected by a sequence of bistellar flips that preserve the vertex colorings.