Haefliger structure

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In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André 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 UαUβ to the sheaf of germs of local diffeomorphisms of , satisfying the 1-cocycle condition

for

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 , such that for each α, β there is a map Φαβ from UαUβ 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.

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 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 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 and concordance classes of Haefliger structures.

References[]

  • Anosov, D.V. (2001) [1994], "Haefliger structure", Encyclopedia of Mathematics, EMS Press
  • Haefliger, André (1970). "Feuilletages sur les variétés ouvertes". Topology. 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. Vol. 197. Berlin, New York: Springer-Verlag. pp. 133–163. doi:10.1007/BFb0068615. MR 0285027.
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