# Haefliger structure

In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970. Any foliation on a manifold induces a special kind of Haefliger structure, which uniquely determines the foliation.

## Definition

A codimension-$q$ Haefliger structure on a topological space $X$ consists of the following data:
* a cover of $X$ by open sets $U_{\alpha }$;
* a collection of continuous maps $f_{\alpha }:X\to \mathbb {R} ^{q}$;
* for every $x\in U_{\alpha }\cap U_{\beta }$, a diffeomorphism $\psi _{\alpha \beta }^{x}$ between open neighbourhoods of $f_{\alpha }(x)$ and $f_{\beta }(x)$ with $\Psi _{\alpha \beta }^{x}\circ f_{\alpha }=f_{\beta }$;
such that the continuous maps $\Psi _{\alpha \beta }:x\mapsto \mathrm {germ} _{x}(\psi _{\alpha \beta }^{x})$ from $U_{\alpha }\cap U_{\beta }$ to the sheaf of germs of local diffeomorphisms of $\mathbb {R} ^{q}$ satisfy the 1-cocycle condition
    $\displaystyle \Psi _{\gamma \alpha }(u)=\Psi _{\gamma \beta }(u)\Psi _{\beta \alpha }(u)$ for $u\in U_{\alpha }\cap U_{\beta }\cap U_{\gamma }.$
The cocycle $\Psi _{\alpha \beta }$ is also called a Haefliger cocycle.
More generally, ${\mathcal {C}}^{r}$, piecewise linear, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.

## Examples and constructions
### Pullbacks

An advantage of Haefliger structures over foliations is that they are closed under pullbacks. More precisely, given a Haefliger structure on $X$, defined by a Haefliger cocycle $\Psi _{\alpha \beta }$, and a continuous map $f:Y\to X$, the pullback Haefliger structure on $Y$ is defined by the open cover $f^{-1}(U_{\alpha })$ and the cocycle $\Psi _{\alpha \beta }\circ f$. As particular cases we obtain the following constructions:
* Given a Haefliger structure on $X$ and a subspace $Y\subseteq X$, the restriction of the Haefliger structure to $Y$ is the pullback Haefliger structure with respect to the inclusion $Y\hookrightarrow X$
* Given a Haefliger structure on $X$ and another space $Y$, the product of the Haefliger structure with $Y$ is the pullback Haefliger structure with respect to the projection $X\times Y\to X$

### Foliations

Recall that a codimension-$q$ foliation on a smooth manifold can be specified by a covering of $X$ by open sets $U_{\alpha }$, together with a submersion $\phi _{\alpha }$ from each open set $U_{\alpha }$ to $\mathbb {R} ^{q}$, such that for each $\alpha ,\beta$ there is a map $\Phi _{\alpha \beta }$ from $U_{\alpha }\cap U_{\beta }$ 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.
As anticipated, foliations are not closed in general under pullbacks but Haefliger structures are. Indeed, given a continuous map $f:X\to Y$, 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 $X\times [0,1]$ to $X\times 0$ and $X\times 1$.
There is a classifying space $B\Gamma _{q}$ 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 $B\Gamma _{q}$ 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 $B\Gamma _{q}$ and concordance classes of Haefliger structures.