In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970.
[1][2] Any foliation on a manifold induces a special kind of Haefliger structure, which uniquely determines the foliation.
Haefliger structure on a topological space
consists of the following data: such that the continuous maps
α β
α β
to the sheaf of germs of local diffeomorphisms of
satisfy the 1-cocycle condition The cocycle
α β
is also called a Haefliger cocycle.
, piecewise linear, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.
An advantage of Haefliger structures over foliations is that they are closed under pullbacks.
More precisely, given a Haefliger structure on
, defined by a Haefliger cocycle
α β
, the pullback Haefliger structure on
is defined by the open cover
As particular cases we obtain the following constructions: Recall that a codimension-
foliation on a smooth manifold can be specified by a covering of
α , β
α β
The Haefliger cocycle is defined by As anticipated, foliations are not closed in general under pullbacks but Haefliger structures are.
is not transverse the pullback can be a Haefliger structure that is not a foliation.
are called concordant if they are the restrictions of Haefliger structures on
Haefliger structures which has a universal Haefliger structure on it in the following sense.
the pullback of the universal Haefliger structure is a Haefliger structure on
For well-behaved topological spaces
this induces a 1:1 correspondence between homotopy classes of maps from
and concordance classes of Haefliger structures.