The Space of Clouds in Euclidean Space

Abstract

We study the space {\small $\nua{m}{d}$} of clouds in {\small $\bbr^d$} (ordered sets of m points modulo the action of the group of affine isometries). We show that {\small $\nua{m}{d}$} is a smooth space, stratified over a certain hyperplane arrangement in {\small $\bbr^m$}. 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 {\small $m\leq 9$} and all the strata when {\small $m\leq 8$}. As the strata are the product of a polygon space with a disk, this gives a classification of {\small $m$}-gon spaces for {\small $m\leq 9$}. When {\small $d=2,3\,$}, {\small $m=5,6,7\,$}, and modulo reordering, we show that the chambers (and so the different generic polygon spaces) are distinguished by the ring structure of their {\small ${\rm mod}\, 2$}-cohomology.