In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by Haefliger (1970, 1971). 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 Uα, together with continuous maps Ψαβ from Uα ∩ Uβ to the sheaf of germs of local diffeomorphisms of Rq, satisfying the 1-cocycle condition
More generally, Cr, PL, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.
Haefliger structure and foliations
A codimension-q foliation can be specified by a covering of X by open sets Uα, together with a submersion φα from each open set Uα to Rq, such that for each α, β there is a map Φαβ from Uα ∩ Uβ to local diffeomorphisms with
whenever v is close enough to u. The Haefliger cocycle is defined by
- germ of at u.
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 BΓq 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 BΓq 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 BΓq and concordance classes of Haefliger structures.
- Anosov, D.V. (2001) , "h/h046120", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Haefliger, André (1970), "Feuilletages sur les variétés ouvertes", Topology. an International Journal of Mathematics, 9: 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, 197, Berlin, New York: Springer-Verlag, pp. 133–163, doi:10.1007/BFb0068615, MR 0285027