  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 $\pi_6(G) \neq 0$, 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.
