Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$ A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back. Let $GH^*(X,Z)$ be the subgroup of $H^*(X,Z)$ formed by the images of the above map $H^*(BG,Z)\to H^*(X,Z)$ for all the maps $X\to BG$.

In this case $a \in GH^*(X,Z)$ and the Stiefel-Whitney/Pontryagin classes may have some relations that are indenependent of $X$. What are those relations?

If we choose $G=U(1)$, this question bacomes: what are the relations (all the $X$ independent relations) between the Chern classes of a $U(1)$ bundle on $X$ and the Stiefel-Whitney/Pontryagin classes on $X$.

In question Relations between Stiefel-Whitney classes, the relations between Stiefel-Whitney classes on any $X$ are discussed.

== Added == This may be the better way to phrase the question: What are relations between the characteristic classes of the tangent bundle of a manifold and characteristic classes of an arbitrary $G$-bundles on the same manifold.

I think in 4-dimension and for $G=U(1)$, one of the relation is $(w_2+w_1^2) c^{U(1)}_1 = c^{U(1)}_1c^{U(1)}_1$ mod 2.

4more comments