Fan's lemma via bistellar moves

TL;DR

This paper extends Pachner's theorem to manifolds with free Z2-actions and provides a combinatorial proof of Fan's lemma for centrally symmetric sphere triangulations, especially in low dimensions.

Contribution

It introduces a Pachner-type theorem for symmetric manifolds and offers a new combinatorial proof of Fan's lemma without extra assumptions in dimensions up to 3.

Findings

Pachner's theorem is extended to Z2-symmetric manifolds.

A combinatorial proof of Fan's lemma is provided for symmetric triangulations.

No additional assumptions are needed for dimensions ≤ 3.

Abstract

Pachner proved that all closed combinatorially equivalent combinatorial manifolds can be transformed into each other by a finite sequence of bistellar moves. We prove an analogue of Pachner's theorem for combinatorial manifolds with a free Z2-action, and use it to give a combinatorial proof of Fan's lemma about labellings of centrally symmetric triangulations of spheres. Similarly to other combinatorial proofs, we must assume an additional property of the triangulation for the proof to work. However, unlike the other combinatorial proofs, no such assumption is needed for dimensions at most 3.