Geodesic
On local combinatorial formulas for Chern classes of a~triangulated circle bundle
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 201-235

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A 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 the combinatorial sense). We express rational local formulas for all powers of the first Chern class in terms of expectations of the parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of a triangulated circle bundle over a simplex, measuring the mixing by the triangulation of the circular graphs over vertices of the simplex. The goal of this note is to sketch the logic of deducing these formulas from Kontsevitch's cyclic invariant connection form on metric polygons. 
@article{ZNSL_2016_448_a13,
     author = {N. Mnev and G. Sharygin},
     title = {On local combinatorial formulas for {Chern} classes of a~triangulated circle bundle},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {201--235},
     publisher = {mathdoc},
     volume = {448},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a13/}
}
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N. Mnev; G. Sharygin. On local combinatorial formulas for Chern classes of a~triangulated circle bundle. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 201-235. http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a13/