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 {\displaystyle q} Haefliger structure on a topological space X {\displaystyle X} consists of the following data:
a cover of X {\displaystyle X} by open sets U α {\displaystyle U_{\alpha }} ; a collection of continuous maps f α : X → R q {\displaystyle f_{\alpha }:X\to \mathbb {R} ^{q}} ; for every x ∈ U α ∩ U β {\displaystyle x\in U_{\alpha }\cap U_{\beta }} , a diffeomorphism ψ α β x {\displaystyle \psi _{\alpha \beta }^{x}} between open neighbourhoods of f α ( x ) {\displaystyle f_{\alpha }(x)} and f β ( x ) {\displaystyle f_{\beta }(x)} with Ψ α β x ∘ f α = f β {\displaystyle \Psi _{\alpha \beta }^{x}\circ f_{\alpha }=f_{\beta }} ; such that the continuous maps Ψ α β : x ↦ g e r m x ( ψ α β x ) {\displaystyle \Psi _{\alpha \beta }:x\mapsto \mathrm {germ} _{x}(\psi _{\alpha \beta }^{x})} from U α ∩ U β {\displaystyle U_{\alpha }\cap U_{\beta }} to the sheaf of germs of local diffeomorphisms of R q {\displaystyle \mathbb {R} ^{q}} satisfy the 1-cocycle condition
Ψ γ α ( u ) = Ψ γ β ( u ) Ψ β α ( u ) {\displaystyle \displaystyle \Psi _{\gamma \alpha }(u)=\Psi _{\gamma \beta }(u)\Psi _{\beta \alpha }(u)} for u ∈ U α ∩ U β ∩ U γ . {\displaystyle u\in U_{\alpha }\cap U_{\beta }\cap U_{\gamma }.}
The cocycle Ψ α β {\displaystyle \Psi _{\alpha \beta }} is also called a Haefliger cocycle. More generally, C r {\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.
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 {\displaystyle X} , defined by a Haefliger cocycle Ψ α β {\displaystyle \Psi _{\alpha \beta }} , and a continuous map f : Y → X {\displaystyle f:Y\to X} , the pullback Haefliger structure on Y {\displaystyle Y} is defined by the open cover f − 1 ( U α ) {\displaystyle f^{-1}(U_{\alpha })} and the cocycle Ψ α β ∘ f {\displaystyle \Psi _{\alpha \beta }\circ f} . As particular cases we obtain the following constructions:
Given a Haefliger structure on X {\displaystyle X} and a subspace Y ⊆ X {\displaystyle Y\subseteq X} , the restriction of the Haefliger structure to Y {\displaystyle Y} is the pullback Haefliger structure with respect to the inclusion Y ↪ X {\displaystyle Y\hookrightarrow X}
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