Haefliger structure

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 U_α, together with continuous maps Ψ_αβ from U_α ∩ U_β to the sheaf of germs of local diffeomorphisms of R^q, 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, C^r, 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 U_α, together with a submersion φ_α from each open set U_α to R^q, such that for each α, β there is a map Φ_αβ from U_α ∩ U_β 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 BΓ_q 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 BΓ_q 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 BΓ_q and concordance classes of Haefliger structures.

References

Anosov, D.V. (2001) [1994], "h/h046120", in Hazewinkel, Michiel, 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

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