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The space of clouds in Euclidean space. (English) Zbl 1053.55015

The authors study ”clouds”, the spaces \(\mathcal{N}_{d}^m\) of equivalence classes of ordered \(m\)-tuples of points in \(\mathbb{R}^d\). Two sets of \(m\)-tuples are equivalent if there is an affine isometry sending one set to the other.
The authors show that \(\mathcal{N}_{d}^m\) is a smooth space, stratified over a certain hyperplane arrangement in \(\mathbb{R}^m\). The arrangement arises from the critical points of the map \(\lambda\) from \(\mathcal{N}_{d}^m\) to \(\mathbb{R}^m\) defined by sending an equivalence class to the \(m\)-tuple of distances to the barycenter of a representative. By virtue of the map \(\lambda\), this stratification defines for each \(a\) in the positive orthant of \(\mathbb{R}^m\) a subset \(\mathcal{N}_{d}^m(a)\). This space can be thought of as a space of \(m\)-gons in \(\mathbb{R}^d\), of “type” \(a\).
The authors investigate the structure of the strata, and the resulting chambers (dimension \(m\) strata). There are many results and their techniques allow new enumerations of the possible strata for some low dimensional cases. One result of many: For \(5\leq m \leq 7\), \(\mathcal{N}_{3}^m(\alpha)\), for distinct chambers \(\alpha\), have non-isomorphic mod \(2\) cohomology rings.

MSC:

55R80 Discriminantal varieties and configuration spaces in algebraic topology
57R19 Algebraic topology on manifolds and differential topology
53D20 Momentum maps; symplectic reduction
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