Le cocycle du verger

We define and prove uniqueness of a natural homomorphism (called the Orchard morphism) from some groups associated naturally to a finite set $E$ to the group $\mathcal{E}(E)$ of two-partitions of $E$ representing equivalence relations having at most two classes on $E$. As an application, given a finite generic configuration ${\mathcal C}\subset {\mathbf R}^d$,weexhibitanaturalpartitionof${\mathcal C}$ in two sets.