  We compute characteristic numbers of crepant resolutions of Weierstrass
models corresponding to elliptically fibered fourfolds $Y$ dual in F-theory to
a gauge theory with gauge group $G$. In contrast to the case of fivefolds,
Chern and Pontryagin numbers of fourfolds are invariant under crepant
birational maps. It follows that Chern and Pontryagin numbers are independent
on a choice of a crepant resolution. We present the results for the Euler
characteristic, the holomorphic genera, the Todd-genus, the $L$-genus, the
$\hat{A}$-genus, and the curvature invariant $X_8$ that appears in M-theory. We
also show that certain characteristic classes are independent on the choice of
the Kodaria fiber characterizing the group $G$. That is the case of $\int_Y
c_1^2 c_2$, the arithmetic genus, and the $\hat{A}$-genus. Thus, it is enough
to know $\int_Y c_2^2$ and the Euler characteristic $\chi(Y)$ to determine all
the Chern numbers of an elliptically fibered fourfold. We consider the cases of
$G=$ SU($n$) for ($n=2,3,4,5,6,7$), USp($4$), Spin($7$), Spin($8$), Spin($10$),
G$_2$, F$_4$, E$_6$, E$_7$, or E$_8$.
