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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 o E M depends only on the ordered star st(or, M). For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class.doi:10.1090/s0002-9947-1978-0494134-6 fatcat:icszz33b4bgcdlkqp7rjz6bjly