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.
Definition
editA 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
editPullbacks
editAn 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
Foliations
editRecall 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.
Classifying space
editTwo 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.
References
edit- Anosov, D.V. (2001) [1994], "Haefliger structure", Encyclopedia of Mathematics, EMS Press
- ^ Haefliger, André (1970). "Feuilletages sur les variétés ouvertes". Topology. 9 (2): 183–194. doi:10.1016/0040-9383(70)90040-6. ISSN 0040-9383. MR 0263104.
- ^ Haefliger, André (1971). "Homotopy and integrability". Manifolds--Amsterdam 1970 (Proc. Nuffic Summer School). Lecture Notes in Mathematics, Vol. 197. Vol. 197. Berlin, New York: Springer-Verlag. pp. 133–163. doi:10.1007/BFb0068615. ISBN 978-3-540-05467-2. MR 0285027.