Computation of characteristic classes of a manifold from a triangulation of it

This paper is devoted to the well-known problem of computing the Stiefel–Whitney classes and the Pontryagin classes of a manifold from a given triangulation of the manifold. In 1940 Whitney found local combinatorial formulae for the Stiefel–Whitney classes. The first combinatorial formula for the first rational Pontryagin class was found by Gabrielov, Gel'fand, and Losik in 1975. Since then, different authors have constructed several different formulae for the rational characteristic classes of a triangulated manifold, but none of these formulae provides an algorithm that computes the characteristic cycle solely from a triangulation of the manifold. In this paper a new local combinatorial formula recently found by the author for the first Pontryagin class is described; it provides the desired algorithm. This result uses a solution of the following problem: construct a function $f$ on the set of isomorphism classes of three-dimensional PL-spheres such that for any combinatorial manifold the chain obtained by taking each simplex of codimension four with coefficient equal to the value of the function on the link of the simplex is a cycle.