Balanced shellings and moves on balanced manifolds

TL;DR

This paper extends classical shelling results to balanced manifolds, proving that balancedness can be maintained during transformations and introducing basic cross-flips as a key tool for connecting PL homeomorphic manifolds.

Contribution

It demonstrates that balanced PL manifolds can be connected via balanced shellings and cross-flips, preserving balancedness and providing enumeration of essential moves.

Findings

Balanced shellings preserve balancedness in manifolds.

Any two balanced PL homeomorphic manifolds can be connected by basic cross-flips.

Approximately half of the basic cross-flips are sufficient for connecting manifolds.

Abstract

A classical result by Pachner states that two $d$-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e., properly $(d + 1)$-colored, manifolds such a sequence can be chosen such that balancedness is preserved in each step. As a key ingredient we establish that any two balanced PL homeomorphic combinatorial manifolds with the same boundary are connected by a sequence of basic cross-flips, as was shown recently by Izmestiev, Klee and Novik for balanced manifolds without boundary. Moreover, we enumerate combinatorially different basic cross-flips and show that roughly half of these suffice to relate any two PL homeomorphic manifolds.