2018/01/09 by Mehrzad Ajoodanian, Ajoodanian, Mehrzad, Eaman Eftekhary +1
Mathematics · Physics and Astronomy · #57R30 #Advanced Topics in Algebra #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1801.02963
openalex publication_date 2018/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.