Characteristic numbers in $U$-theory

We compute the subgroup of $\Omega_*^U\otimes Q$ consisting of all elements with integral $U$-numbers. Using these results, we obtain a new computation of the group $\mathrm{Ext}^1_{A^U}(\Omega_*^U,\Omega_*^U)$ and a complete answer to the question of Chern numbers of $(U,\mathrm{fr})$-manifolds.