We present a local combinatorial formula for Euler class of $n$-dimensional PL spherical fiber bundle as a rational number $e_{it CH}$ associated to chain of $n+1$ abstract subdivisions of abstract $n$-spherical PL cell complexes. The number $e_{it C
H}$ is combinatorial (or matrix) Hodge theory twisting cochain in Guy Hirshs homology model of the bundle associated with PL combinatorics of the bundle.
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.
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 necklac
e (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 Kontsevitchs cyclic invariant connection form on metric polygons.
Pure combinatorial models for BPL_n and Gauss map of a combinatorial manifold are described.
