We perform a bordism computation to show that the $E_{7(7)}(\mathbb{R})$
U-duality symmetry of 4d $\mathcal N = 8$ supergravity could have an anomaly
invisible to perturbative methods; then we show that this anomaly is trivial.
We compute the relevant bordism group using the Adams and Atiyah-Hirzebruch
spectral sequences, and we show the anomaly vanishes by computing
$\eta$-invariants on the Wu manifold, which generates the bordism group.
