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Haefliger structure - Wikipedia

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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- $q$ Haefliger structure on a topological space $X$ consists of the following data:

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

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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:

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.

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.

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

^ Haefliger, André (1970). "Feuilletages sur les variétés ouvertes". Topology. 9 (2): 183–194. doi:10.1016/0040-9383(70)90040-6. ISSN 0040-9383. MR 0263104.
^ Haefliger, André (1971). "Homotopy and integrability". Manifolds--Amsterdam 1970 (Proc. Nuffic Summer School). Lecture Notes in Mathematics, Vol. 197. Vol. 197. Berlin, New York: Springer-Verlag. pp. 133–163. doi:10.1007/BFb0068615. ISBN 978-3-540-05467-2. MR 0285027.