ИСТИНА

In 1958 R.Thom showed that Pontrjagin classes of a triangulated manifold can be determined from combinatoric data, encoded by its triangulation. Later N.Levitt and C.P. Rourke showed, that there exist local formulas, that represent simplicial cocycles, corresponding to this classes, in terms of the links of simplices in this triangulation. However, these results are pure existence theorems, and the problem to find explicit expressions for such classes is still open (one should mention papers by Gelfand, Gabrielov, Losik, Gelfand and MaPherson, and Gaifullin, dealing with it). In my talk I am going to describe a solution of a similar, but simpler problem: suppose we have a triangulated principal $S^1$-bundle; I shall give explicit local formulas to express powers of its Chern class in terms of this triangulation. One can use this result to evaluate the number of simplices, necessary to triangulate a base of an $S^1$-bundle so that there will exist a triangulation of the total space, that agrees with it. The talk is based on a joint work with N.Mn\"ev (POMI).