Generalization of a foliation
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.
A codimension- Haefliger structure on a topological space consists of the following data:
- a cover of by open sets ;
- a collection of continuous maps ;
- for every , a diffeomorphism between open neighbourhoods of and with ;
such that the continuous maps from to the sheaf of germs of local diffeomorphisms of satisfy the 1-cocycle condition
- for
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
[edit]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 , and a continuous map , the pullback Haefliger structure on is defined by the open cover and the cocycle . As particular cases we obtain the following constructions:
- Given a Haefliger structure on and a subspace , the restriction of the Haefliger structure to is the pullback Haefliger structure with respect to the inclusion
- Given a Haefliger structure on and another space , the product of the Haefliger structure with is the pullback Haefliger structure with respect to the projection
Recall that a codimension- foliation on a smooth manifold can be specified by a covering of by open sets , together with a submersion from each open set to , such that for each there is a map from to local diffeomorphisms with
whenever is close enough to . The Haefliger cocycle is defined by
- germ of at u.
As anticipated, foliations are not closed in general under pullbacks but Haefliger structures are. Indeed, given a continuous map , one can take pullbacks of foliations on provided that is transverse to the foliation, but if is not transverse the pullback can be a Haefliger structure that is not a foliation.
Two Haefliger structures on are called concordant if they are the restrictions of Haefliger structures on to and .
There is a classifying space for codimension- Haefliger structures which has a universal Haefliger structure on it in the following sense. For any topological space and continuous map from to 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 to and concordance classes of Haefliger structures.