On vanishing of characteristic numbers in Poincaré complexes

Abstract:

Let $G_r(X)\subset \pi _r(X)$ be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map $G_r(X)\rightarrow H_r(X; R)$ is non-trivial and $R$ is a field. We will prove: if $X$ is a Poincaré complex oriented in $R$-coefficient, all the characteristic numbers of $X$ in $R$-coefficient vanish. Similarly, if $R=Z_p$ and $X$ is a $Z_p$-Poincaré 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.