The Existence of Combinatorial Formulae for Characteristic Classes
Norman Levitt, Colin Rourk · Transactions of the American Mathematical Society · 1978
Given a characteristic class on a locally ordered combinatorial manifold M there exists a cocycle which represents the class on M and is locally defined, i.e. its value on $\sigma \in M$ depends only on the ordered star ${\text {st}}(\sigma ,M)$. For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class.