Gauge theory and foliations I; germ cords versus quantum cords

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We apply gauge theory to study the space Fₖ(M) of smooth codimension-k framed foliations on a smooth manifold M. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains Fₖ(M) as a subspace. The notion of holonomy is naturally extended to this moduli space and the cohomology theory associated with points of this moduli space which correspond to non-singular foliations coincides with Bott cohomology. The quotient of the moduli space under concordance is identified as the space of homotopy classes of maps to the classifying spaces BΓᵍₖ and BΓqₖ. While BΓᵍ is a classic and has been studied since Haefliger, BΓq (which is a quotient of BΓᵍ) carries a simpler topology and offers a rival theory.