M5 brane to D4 brane via cyclification of rational relative 3-cohomotopy

In this article, we start by re-deriving the equations of motion and Bianchi identities for the abelian D4 brane worldvolume. Noting that the 4-flux in M-Theory is rationally flux quantized in a non-abelian cohomology theory called 4-cohomotopy, and the three-flux on the M5 brane worldvolume in rational (twisted) 3-cohomotopy, we compute the minimal model for the cyclification of the quaternionic Hopf fibration which encodes the Bianchi identities for the fluxes on the D4 brane worldvolume after double dimensional reduction. The two pictures can be mapped to each other, and thus at the rational level, we conjecture a non-abelian relative cohomology theory for the D4 brane, fibered over the 10d Type IIA spacetime fluxes.