We define and study a generalized Wang homomorphism

closely related to the generalized Gottlieb group. 

We show that the existence of an element in the 

generalized Gottlieb group whose image under the 

Hurewicz map is nonzero is a sufficient condition 

for the vanishing of the mod $p$ Wu numbers of $X$.

Also, we can show that the image 

$\lambda _{\hat\alpha }^q(k)$ of $k$ under the 

generalized Wang homomorphism is an obstruction to a

map $S^n\times E_{kf}\to X$ lifting to a map 

$S^n\times E_{kf}\to E_k$, where $E_k$ is the 

fibration induced by $k\in H^q(X;\pi )$