In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970. Any foliation on a manifold induces a special kind of Haefliger structure, which uniquely determines the foliation.

Definition

A codimension-q Haefliger structure on a topological space X consists of the following data: such that the continuous maps from to the sheaf of germs of local diffeomorphisms of \R^q satisfy the 1-cocycle condition The cocycle **' is also called a **Haefliger cocycle'''. More generally, , piecewise linear, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.

Examples and constructions

Pullbacks

An advantage of Haefliger structures over foliations is that they are closed under pullbacks. More precisely, given a Haefliger structure on X, defined by a Haefliger cocycle ', and a continuous map f: Y \to X, the pullback Haefliger structure on Y is defined by the open cover and the cocycle '. As particular cases we obtain the following constructions:

Foliations

Recall that a codimension-q foliation on a smooth manifold can be specified by a covering of X by open sets U_\alpha, together with a submersion \phi_\alpha from each open set U_\alpha to \R^q, such that for each there is a map from to local diffeomorphisms with whenever v is close enough to u. The Haefliger cocycle is defined by As anticipated, foliations are not closed in general under pullbacks but Haefliger structures are. Indeed, given a continuous map f: X \to Y, 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.

Classifying space

Two Haefliger structures on X are called concordant if they are the restrictions of Haefliger structures on to X \times 0 and X \times 1. There is a classifying space B\Gamma_q for codimension-q Haefliger structures which has a universal Haefliger structure on it in the following sense. For any topological space X and continuous map from X to B\Gamma_q the pullback of the universal Haefliger structure is a Haefliger structure on X. For well-behaved topological spaces X this induces a 1:1 correspondence between homotopy classes of maps from X to B\Gamma_q and concordance classes of Haefliger structures.