Haefliger structure

Remarks

The cocycle condition with $i = j = k$ shows that $\gamma_{ii}(x)$ is the germ of the identity at a point $f_i(x) \in \R^q$, and this defines continuous maps $f_i \colon U_i \to \R^q$. The germ $\gamma_{ij}(x)$ goes from $f_j(x)$ to $f_i(x)$, so near $x$ the local maps satisfy $f_i = \phi \circ f_j$ for a representative $\phi$. The same definition works with any topological groupoid in place of $\Gamma_q$.

Example: let $\mathcal{F}$ be a foliation of codimension $q$ of a manifold $X$, given by submersions $f_i \colon U_i \to \R^q$ with $f_i = g_{ij} \circ f_j$ on overlaps for local diffeomorphisms $g_{ij}$. Taking $\gamma_{ij}(x)$ to be the germ of $g_{ij}$ at $f_j(x)$ gives a $\Gamma_q$-cocycle, since the transition maps compose as $g_{ik} = g_{ij} \circ g_{jk}$ near each point. Another example: any continuous map $f \colon X \to \R^q$ gives a $\Gamma_q$-structure with one open set $U_1 = X$ and $\gamma_{11}(x)$ the germ of the identity at $f(x)$.

Non-example: let $X$ be a point, $U_1 = U_2 = X$, $\gamma_{11} = \gamma_{22} = \gamma_{21}$ the germ of the identity at $0$ and $\gamma_{12}$ the germ of $t \mapsto 2t$ at $0$. The cocycle condition fails for $(i, j, k) = (1, 2, 1)$, since $\gamma_{12} \gamma_{21}$ is the germ of $t \mapsto 2t$ and not that of the identity.

Haefliger structures pull back along continuous maps. For a CW complex $X$, homotopy classes of $\Gamma_q$-structures on $X$ correspond to homotopy classes of maps from $X$ into a classifying space $B\Gamma_q$.