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.
