Haefliger structure

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 ${\displaystyle \mathbb {R} ^{q}}$, satisfying the 1-cocycle condition

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

More generally, C^r, PL, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.

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 ${\displaystyle \mathbb {R} ^{q}}$, such that for each α, β there is a map Φ_αβ from U_α ∩ U_β to local diffeomorphisms with

${\displaystyle \phi _{\alpha }(v)=\Phi _{\alpha ,\beta }(u)(\phi _{\beta }(v))}$

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

${\displaystyle \Psi _{\alpha ,\beta }(u)=}$ germ of ${\displaystyle \Phi _{\alpha ,\beta }(u)}$ 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.

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 ${\displaystyle B\Gamma _{q}}$ 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 ${\displaystyle B\Gamma _{q}}$ 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 ${\displaystyle B\Gamma _{q}}$ and concordance classes of Haefliger structures.

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• 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.