Combining ideas from Whitney’s geometric integration theory and rough analysis, we

introduce spaces of rough differential k–forms on d-manifolds which are formally given by

f = P

I fI dxI where (fI )I belong to a class of genuine distributions of negative regularity.

These rough k–forms have several properties desirable of a notion of differential forms:

• they can be integrated over suitably regular k-manifolds,

• they form a module under point-wise multiplication with sufficiently regular functions,

• exterior differentiation as well as the Stokes theorem extend to these spaces,

• they come with natural embeddings into distribution spaces,

• they contain classes of form valued distributional random fields.

Finally, these spaces unify several previous constructions in the literature. In particular, they

generalise spaces of α–flat cochains introduced by Whitney and Harrison, they contain the

(rough) k-forms f · dg1 ∧ ... ∧ dgk introduced by Z¨ust using Young integration, and for d = 2

and k = 1, they are close to the spaces which Chevyrev et al. use to make sense of Yang–

Mills connections. Lastly, as a technical tool we introduce a ‘simplicial sewing lemma’, which

provides a coordinate invariant formulation of the (known) multi-dimensional sewing lemma.