In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970.^[1][2] Any foliation on a manifold induces a special kind of Haefliger structure, which uniquely determines the foliation.

A codimension-${\displaystyle q}$ Haefliger structure on a topological space ${\displaystyle X}$ consists of the following data:

• a cover of ${\displaystyle X}$ by open sets ${\displaystyle U_{\alpha }}$;
• a collection of continuous maps ${\displaystyle f_{\alpha }:X\to \mathbb {R} ^{q}}$;
• for every ${\displaystyle x\in U_{\alpha }\cap U_{\beta }}$, a diffeomorphism ${\displaystyle \psi _{\alpha \beta }^{x}}$ between open neighbourhoods of ${\displaystyle f_{\alpha }(x)}$ and ${\displaystyle f_{\beta }(x)}$ with ${\displaystyle \Psi _{\alpha \beta }^{x}\circ f_{\alpha }=f_{\beta }}$;

such that the continuous maps ${\displaystyle \Psi _{\alpha \beta }:x\mapsto \mathrm {germ} _{x}(\psi _{\alpha \beta }^{x})}$ from ${\displaystyle U_{\alpha }\cap U_{\beta }}$ to the sheaf of germs of local diffeomorphisms of ${\displaystyle \mathbb {R} ^{q}}$ satisfy the 1-cocycle condition

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

The cocycle ${\displaystyle \Psi _{\alpha \beta }}$ is also called a Haefliger cocycle.

More generally, ${\displaystyle {\mathcal {C}}^{r}}$, piecewise linear, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.

Two Haefliger structures on ${\displaystyle X}$ are called concordant if they are the restrictions of Haefliger structures on ${\displaystyle X\times [0,1]}$ to ${\displaystyle X\times 0}$ and ${\displaystyle X\times 1}$.

There is a classifying space ${\displaystyle B\Gamma _{q}}$ for codimension-${\displaystyle q}$ Haefliger structures which has a universal Haefliger structure on it in the following sense. For any topological space ${\displaystyle X}$ and continuous map from ${\displaystyle X}$ to ${\displaystyle B\Gamma _{q}}$ the pullback of the universal Haefliger structure is a Haefliger structure on ${\displaystyle X}$. For well-behaved topological spaces ${\displaystyle X}$ this induces a 1:1 correspondence between homotopy classes of maps from ${\displaystyle X}$ to ${\displaystyle B\Gamma _{q}}$ and concordance classes of Haefliger structures.