Relation between triangular- and Witten-SU(2)-anomaly cancellation for gauge groups

We show by group-theoretic means that the absence of the Witten SU(2) (global) anomaly for gauge groups is guaranteed by the (perturbative) triangular anomaly-free condition as long as the SU(2) gauge group is embedded in a larger simple gauge group $G$ with the property ${\Pi }_4\mathrm{}\left(G\right)=0\mathrm{}$ (${\Pi }_4$ is the four-dimensional homotopy group). The inverse is not true. The same method could be used to relate perturbative and global anomaly cancellation in more than four dimensions.