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2004 The Space of Clouds in Euclidean Space
Jean-Claude Hausmann, Eugenio Rodriguez
Experiment. Math. 13(1): 31-48 (2004).


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.


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Jean-Claude Hausmann. Eugenio Rodriguez. "The Space of Clouds in Euclidean Space." Experiment. Math. 13 (1) 31 - 48, 2004.


Published: 2004
First available in Project Euclid: 10 June 2004

zbMATH: 1053.55015
MathSciNet: MR2065566

Primary: 55R80 , 57R1
Secondary: 53D20

Keywords: configuration spaces , Polygon spaces

Rights: Copyright © 2004 A K Peters, Ltd.

Vol.13 • No. 1 • 2004
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