We cast M-brane interactions including intersecting membranes and five-branes in manifestly gauge invariant form using an arrangement of higher dimensional Dirac surfaces. We show that the noncommutative gauge symmetry present in the doubled M-theory formalism involving dual 3-form and 6-form gauge fields is preserved in a form quantised over the integers. The proper context for discussing large noncommutative gauge transformations is relative cohomology, in which the 3-form transformation parameters become exact when restricted to the five-brane worldvolume. We show how this structure yields the lattice of M-theory charges and gives rise to the conjectured 7D Hopf–Wess–Zumino term.