Cohomotopy and flux quantization in $M$-theory

We identify some of the $k$-invariants for the Postnikov tower of the stable and unstable 4-sphere. Assuming the stable Hypothesis H of Fiorenza–Sati–Schreiber, we use the resulting obstruction theory to prove that the Chern–Simons term in the effective action of M-theory is well defined. In particular, we do not assume the presence of an $E_8$-gauge field.