Local formulae for combinatorial Pontryagin classes

A. A. Gaifullin

Izvestiya. Mathematics , Tome 68 (2004) no. 5, pp. 861-910

Résumé

Let $p(|K|)$ be the characteristic class of a combinatorial manifold $K$ given by a polynomial $p$ in the rational Pontryagin classes of $K$. We prove that for any polynomial $p$ there is a function taking each combinatorial manifold $K$ to a cycle $z_p(K)$ in its rational simplicial chains such that: 1) the Poincaré dual of $z_p(K)$ represents the cohomology class $p(|K|)$; 2) the coefficient of each simplex $\Delta$ in the cycle $z_p(K)$ is determined solely by the combinatorial type of $\operatorname{link}\Delta$. We explicitly describe all such functions for the first Pontryagin class. We obtain estimates for the denominators of the coefficients of the simplices in the cycles $z_p(K)$.