A bistellar flip on a triangulated manifold is a local move that changes the combinatorial structure of the triangulation without changing its topological homeomorphism type. A classical result in combinatorial topology states that two piecewise-linear (PL) manifold triangulations are PL homeomorphic if and only if they can be related through a series of bistellar flips (one should view the PL condition as a sort of smoothness condition on both the manifold and its triangulation).

A balanced simplicial complex of dimension $d$ is one whose underlying graph admits a proper $(d+1)$-coloring. In general, a bistellar flip may destroy the balanced condition. In this talk, we will introduce a sequence of alternate moves, called cross-flips, which preserve both the balanced condition and homeomorphism type. Hence, these moves act as an analogue of bistellar flips for balanced manifold triangulations.