Pseudosphere arrangements with simple complements (Q1880854)

A pseudosphere arrangement \({\mathcal A}=\{A_1,\ldots, A_k\}\) in a topological \(n\)-sphere \(S^n\) is a finite collection of closed subsets of \(S^n\) such that each nonempty intersection of sets in \(\mathcal A\), \(\bigcap A_I:= \bigcap_{i\in I}A_i\), is either a single point or homeomorphic to a sphere of some dimension. If \(A_I\) is a topological sphere of dimension \(r\geq -1\), the author defines the degeneracy index \(d_I = r-(n-| I| )\), and if \(A_I\) is a point, the author sets \(d_I =\infty\). A pseudosphere arrangement in \(S^n\) is complement simple if each nonempty component of the complement space has the homology of a point. A pseudosphere arrangement in \(S^n\) is fully complement simple if all subarrangements are complement simple. The author proves the following main result: The pseudosphere arrangement \(\mathcal A\) is fully complement simple if and only if \(d_I \geq 0\) for all \(I \subseteq\{1,\dots, k\}\). As a corollary of this result, the author provides a formula for counting the complement components, generalizing [\textit{T. Zaslalvsky}, Mem. Am. Math. Soc. 154 (1975; Zbl 0296.50010)].