Haefliger Structure

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.

Pullbacks

An advantage of Haefliger structures over foliations is that they are closed under pullbacks. More precisely, given a Haefliger structure on ${\displaystyle X}$ , defined by a Haefliger cocycle ${\displaystyle \Psi _{\alpha \beta }}$ , and a continuous map ${\displaystyle f:Y\to X}$ , the pullback Haefliger structure on ${\displaystyle Y}$  is defined by the open cover ${\displaystyle f^{-1}(U_{\alpha })}$  and the cocycle ${\displaystyle \Psi _{\alpha \beta }\circ f}$ . As particular cases we obtain the following constructions:

• Given a Haefliger structure on ${\displaystyle X}$  and a subspace ${\displaystyle Y\subseteq X}$ , the restriction of the Haefliger structure to ${\displaystyle Y}$  is the pullback Haefliger structure with respect to the inclusion ${\displaystyle Y\hookrightarrow X}$ 
• Given a Haefliger structure on ${\displaystyle X}$  and another space ${\displaystyle Y}$ , the product of the Haefliger structure with ${\displaystyle Y}$  is the pullback Haefliger structure with respect to the projection ${\displaystyle X\times Y\to X}$ 

Foliations

Recall that a codimension-${\displaystyle q}$  foliation on a smooth manifold can be specified by a covering of ${\displaystyle X}$  by open sets ${\displaystyle U_{\alpha }}$ , together with a submersion ${\displaystyle \phi _{\alpha }}$  from each open set ${\displaystyle U_{\alpha }}$  to ${\displaystyle \mathbb {R} ^{q}}$ , such that for each ${\displaystyle \alpha ,\beta }$  there is a map ${\displaystyle \Phi _{\alpha \beta }}$  from ${\displaystyle U_{\alpha }\cap U_{\beta }}$  to local diffeomorphisms with

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

whenever ${\displaystyle v}$  is close enough to ${\displaystyle u}$ . The Haefliger cocycle is defined by

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

As anticipated, foliations are not closed in general under pullbacks but Haefliger structures are. Indeed, given a continuous map ${\displaystyle f:X\to Y}$ , one can take pullbacks of foliations on ${\displaystyle Y}$  provided that ${\displaystyle f}$  is transverse to the foliation, but if ${\displaystyle f}$  is not transverse the pullback can be a Haefliger structure that is not a foliation.

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.

• Anosov, D.V. (2001) [1994], "Haefliger structure", Encyclopedia of Mathematics, EMS Press