Local formulae for combinatorial Pontrjagin classes

Abstract

By p(|K|) denote the characteristic class of a combinatorial manifold K given by the polynomial p in Pontrjagin classes of K. We prove that for any polynomial p there exists a function taking each combinatorial manifold K to a rational simplicial cycle z(K) such that: (1) the Poincare dual of z(K) represents the cohomology class p(|K|); (2) a coefficient of each simplex in the cycle z(K) is determined only by the combinatorial type of the link of this simplex. We also prove that if a function z satisfies the condition (2), then this function automatically satisfies the condition (1) for some polynomial p. We describe explicitly all such functions z for the first Pontrjagin class. We obtain estimates for denominators of coefficients of simplices in the cycles z(K).