A local combinatorial formula for the Chern class of a triangulated $S^1$ bundle in terms of shellings

TL;DR

This paper presents a simple combinatorial formula for the first Chern class of a triangulated $S^1$ bundle over a 2-simplex, using shellings and cyclic words to facilitate easy computation.

Contribution

It introduces a local combinatorial formula for the Chern class based on shellings and cyclic words, simplifying calculations for triangulated bundles.

Findings

Provides a rational combinatorial characteristic called 'curvature'

Expresses the curvature in terms of cyclic words

Enables straightforward computation of the Chern class from triangulation

Abstract

Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") of a triangulated $S^1$ bundle over a 2-simplex, which is a local combinatorial formula for the first Chern class. The curvature is expressed in terms of cyclic word in 3-character alphabet associated to the bundle. From the point of view of simplicial combinatorics the word is a canonical shelling of the total complex. If you know a triangulation of a bundle - you can really easily compute the Chern class.