On local combinatorial formulas for Euler class of spherical fiber bundle

We will discuss the classical problem on local combinatorial formulas of characteristic classes on example of Euler class. Suppose we have a PL spherical fiber bundle with a fiber S^n triangulated over the base simplicial complex. The bundle determines n+1 dimensional Euler characteristic class in the base. Local combinatorial formula for the Euler class is a universal combinatorial function of elementary triangulated S^n-bundles over n+1 simplices universally representing Euler cocycle of the bundle in simplicial cohomology of the base. Such functions exist for rational coefficients in cohomology. They can be constructed as "twisting cochains" – explicit local chain-level formulas for Gysin homomorphism in the Gysin sequence of the bundle. To get an access to local chain combinatorics of spectral sequence of the bundle we may use Guy Hirsh homology model of the bundle as a local system and then applying homology perturbation theory obtain local formulas as certain measure of twisting in combinatorial Hodge structure of the elementary bundle. The answer can be interpreted and evaluated statistically as certain combinatorial counting using Catanzaro-Chernyak-Klein higher Kirchhoff theorems. The formulas are resulting in a combinatorial form of Gauss-Bonne theorem. For example we easily obtain otherwise difficult to access statement: One can triangulate only trivial and Hopf circle bundles over a 2-dimensional sphere if the base sphere is triangulated as the boundary of 3-simplex.