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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.

Definition

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

\( \displaystyle\Psi_{\gamma\alpha}(u) = \Psi_{\gamma\beta}(u)\Psi_{\beta\alpha}(u) for u\in U_\alpha\cap U_\beta\cap U_\gamma. \)

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 UαUβ to local diffeomorphisms with

\( \phi_\alpha(v)= \Phi_{\alpha,\beta}(u)(\phi_\beta(v)) \)

whenever v is close enough to u. The Haefliger cocycle is defined by

\( \Psi_{\alpha,\beta}(u) = germ of \Phi_{\alpha,\beta}(u) at u. \)

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.


Classifying space

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.


References

Anosov, D.V. (2001), "h/h046120", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, 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

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