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Formulas for characteristic classes of combinatoric bundles

Le : 17/02/2015 14h30
Par : Georgi Sharygin (Université d'Etat de Moscou)
Lieu : I 001
Lien web :
Résumé : As one knows, every complex or real oriented vector bundle over a CW complex gives rise to a series of classes in the cohomology groups of this complex. These classes are called Chern or Pontryagin classes of the bundle. These classes are important invariants of the bundle and have numerous fine properties and applications; for example, rational Pontryagin classes of the tangent bundle of a manifold allow one classify the manifolds up to a rational cobordism. Suppose now, that the base of the bundle is triangulated; consider the spherical bundle, associated with it. One can show, that this spherical bundle can always be triangulated in a manner, consistent with the triangulation of the base. Then one can ask if it is possible to (algorithmically) find the simplicial cocycles, representing characteristic classes from the combinatoric information, provided by the given triangulations? Since the beginning of 1970s it is known that such formulas exist; however explicit constructions of these formulas has been the subject of research till the present day. In my talk, after a brief excursion into the history of this problem, I will describe major results and ideas of our joint work with N.Mnëv, POMI (St.Petersburgh). First of all, I will give a very explicit description of characteristic classes in the simplest case of 1-dimensional simplicial spherical bundles, i.e. circle bundles. It is not so difficult to find an explicit formula for the first (and the only) Chern class in this case. Rather unexpectedly, similar simple formulas can be given for the powers of this class; an important role in this construction is played by a combinatoric connection on the space of metric polygons, introduced by Kontsevich in a totally different setting. One immediate application of this construction is a description of triangulations of circle bundles over two-dimensional surfaces. In the general case, however, such simple approach can hardly yield very much. In its stead we suggest using some constructions of simplicial sheaves, twisting cochains, and a variant of Igusa-Klein higher Franz-Reidemeister torsion. I will try to explain the reasons we have for thinking so.