A local combinatorial formula for the Chern class of a triangulated $S^1$ bundle in terms of shellings

Details A local combinatorial formula for the Chern class of a triangulated $S^1$ bundle in terms of shellings A local combinatorial formula for the Chern class of a triangulated $S^1$ bundle in terms of shellings

2011-08-24

arxiv.org/abs/1108.4733v2

Mathematics

Algebraic Topology

10 pages 5 figures. In the last version some conjectures and thanks are added

Scientific paper

Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") of a triangulated $S^1$ bundle over a 2-simplex, which is a local combinatorial formula for the first Chern class. The curvature is expressed in terms of cyclic word in 3-character alphabet associated to the bundle. From the point of view of simplicial combinatorics the word is a canonical shelling of the total complex. If you know a triangulation of a bundle - you can really easily compute the Chern class.

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