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.