Haefliger structure  In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by . Any foliation on a manifold induces a Haefliger structure, which uniquely determines the foliation. A Haefliger structure on a space "X" is determined by a Haefliger cocycle. A codimension-"q" Haefliger cocycle consists of a covering of "X" by open sets "U"α, together with continuous maps Ψαβ from  "U"α ∩ "U"β to the sheaf of germs of local diffeomorphisms of formula_1, satisfying the 1-cocycle condition  More generally, "C""r", PL, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves. A codimension-"q" foliation can be specified by a covering of "X" by open sets "U"α, together with a submersion φα from each open set "U"α to formula_1, such that  for each α, β there is a map Φαβ from  "U"α ∩ "U"β to local diffeomorphisms with  whenever "v" is close enough to "u". The Haefliger cocycle is defined by  An advantage of Haefliger structures over foliations is that they are closed under pullbacks. If "f" is a continuous map from "X" to "Y" then one can take pullbacks of foliations on "Y" provided that "f" is transverse to the foliation, but if "f" is not transverse the pullback can be a Haefliger structure that is not a foliation. Two Haefliger structures on "X" are called concordant if they are the restrictions of Haefliger structures on  "X"×[0,1] to "X"×0 and "X"×1. If "f" is a continuous map from "X" to "Y", then there is a pullback under "f" of Haefliger structures on "Y" to Haefliger structures on "X". There is a classifying space formula_8 for codimension-"q" Haefliger structures which has a universal Haefliger structure on it in the following sense.