Local formulae for combinatorial Pontrjagin classes

TL;DR

This paper develops explicit local formulae for combinatorial Pontrjagin classes, linking combinatorial structures to topological invariants through rational cycles determined by local link types.

Contribution

It proves the existence of local formulas for all polynomial Pontrjagin classes and explicitly describes these formulas for the first Pontrjagin class, including coefficient estimates.

Findings

Existence of local formulas for combinatorial Pontrjagin classes.

Explicit description of formulas for the first Pontrjagin class.

Bounds on denominators of cycle coefficients.

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).