On local combinatorial formulas for Chern classes of triangulated circle bundle
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.
Keywords
Cite
@article{arxiv.1608.04708,
title = {On local combinatorial formulas for Chern classes of triangulated circle bundle},
author = {Nikolai Mnev and Georgy Sharygin},
journal= {arXiv preprint arXiv:1608.04708},
year = {2016}
}
Comments
After submitting this note we discovered that the main computation in Sections 5,6 is equivalent to computation in Sections 1,2 http://arxiv.org/abs/math/0207042 of universal combinatorial cochains on cyclic category out of universal cyclic connection form. Still we have an accent on geometry and combinatorics of triangulations