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Regions cut by arrangements of topological spheres. (English) Zbl 0799.52010

An arrangement \(A\) of pseudohyperplanes in \(\mathbb{R}^ n\) is the image, under stereographic projection, of a collection of proper subsets \(A_ 1, \dots, A_ k\) of a topological \(n\)-sphere \(S^ n\) having spherical intersections such that
(i) each \(A_ i\) is a topological \((n-1)\)-sphere containing the pole \(N\),
(ii) for \(I \subseteq \{1, \dots, k\}\), \(I \neq \emptyset\), either \(\cap_{i \in I} (A_ i) = \{N\}\) or \(\cap_{i \in I} (A_ i)\) is a topological sphere of dimension \(\geq n - | I | \geq 0\).
By showing that the complement components are homologically trivial, the author extends the Wiener-Zaslavsky formula on the number of regions induced by \(A\) in terms of intersection degeneracies.

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
51M20 Polyhedra and polytopes; regular figures, division of spaces
05A99 Enumerative combinatorics
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