On local combinatorial formulas for Chern classes of triangulated circle bundle

TL;DR

This paper develops combinatorial formulas for the Chern classes of triangulated circle bundles over polyhedra, linking topology with combinatorics through necklaces and cyclic invariants.

Contribution

It introduces rational local formulas for Chern classes based on combinatorial invariants derived from necklaces associated with triangulated circle bundles.

Findings

Rational formulas for powers of the first Chern class expressed via combinatorial expectations.

Necklaces serve as combinatorial invariants measuring bundle complexity.

Connections to Kontsevich's cyclic invariant form provide a topological foundation.

Abstract

Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all powers of first Chern class in the terms of mathematical expectations of parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of triangulated circle bundle over simplex, measuring mixing by triangulation of the circular graphs over vertices of the simplex. The goal of this note is to sketch the logic of deduction these formulas from Kontsevitch's cyclic invariant connection form on metric polygons.