# Minimal triangulations of circle bundles, circular permutations and binary Chern cocycle

## TL;DR

This paper explores which circle bundles can be triangulated over a base triangulation, highlighting the role of minimal triangulations and circular permutations, and introduces a universal binary Chern cocycle related to cyclic orders.

## Contribution

It provides a new characterization of triangulable circle bundles using local systems of circular permutations and the binary Chern cocycle.

## Key findings

- Minimal triangulations are characterized by local systems of circular permutations.
- The classical Huntington transitivity axiom is expressed as a universal binary Chern cocycle.
- The approach links PL topology with algebraic structures like cocycles.

## Abstract

We investigate a PL topology question: which circle bundles can be triangulated over a given triangulation of the base? The question got a simple answer emphasizing the role of minimal triangulations encoded by local systems of circular permutations of vertices of the base simplices. The answer is based on an experimental fact: classical Huntington transitivity axiom for cyclic orders can be expressed as the universal binary Chern cocycle.