Omega vs. pi, and 6d anomaly cancellation

In this note we review the role of homotopy groups in determining non-perturbative (henceforth `global) gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of $pi_d(G)$ is neither a necessary nor a sufficient condition for there being a possible global anomaly in a $d$-dimensional chiral gauge theory with gauge group $G$. To showcase the failure of sufficiency, we revisit `global anomalies that have been previously studied in 6d gauge theories with $G=SU(2)$, $SU(3)$, or $G_2$. Even though $pi_6(G) eq 0$, the bordism groups $Omega_7^mathrm{Spin}(BG)$ vanish in all three cases, implying there are no global anomalies. In the case of $G=SU(2)$ we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.