On vanishing of characteristic numbers in Poincare complexes

Let G(tau)(X) subset of pi(tau)(X) be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map G(tau)(X) --> H-tau(X; R) is non-trivial and R is a field. We will prove: if X is a Poincare complex oriented in R-coefficient, all the characteristic numbers of X in R-coefficient vanish. Similarly, if R = Z(p) and X is a Z(p)-Poincare complex, then all the mod p Wu numbers vanish. We will also show that the existence of a non-trivial derivation on H*(X; Z(p)) with some suitable conditions implies vanishing of mod p Wu numbers.