The space of clouds in an Euclidean space Jean - Claude HAUSMANN and Eugenio RODRIGUEZ

We study the space Nm d of clouds in Rd (ordered sets of m points modulo the action of the group of affine isometries). We show that Nm d is a smooth space, stratified over a certain hyperplane arrangement in Rm. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for m ≤ 9 and all the strata when m ≤ 8. As the strata are the product of a polygon spaces with a disk, this gives a classification of m-gon spaces for m ≤ 9. When d = 2, 3, m = 5, 6, 7 and modulo reordering, we show that the chambers (and so the different generic polygon spaces) are distinguished by the ring structure of their mod 2-cohomology.