Intersection properties for coverings of \(G\)-spaces
Let \(X\) be a \(G\)-space, and let \(G^*\) denote \(G\setminus \{e\}\). An open subset \(U\) of \(X\) is a \textit{color} if \(U\cap gU=\emptyset\) for all \(g\in G^*\). The \textit{genus} \(\text{gen}(X,G)\) is the minimum cardinality of a covering of \(X\) by sets of the form \(GU\), where \(U\) is a color. In addition, if \(\text{gen}(X,G)=k\) and \(m \geq k\), then a collection \(\{V_i : i\leq m\}\) of subsets of \(G\) is an \textit{intersection family} for \(X\) if for every choice of colors \(\{U_i : i\leq m\}\) such that the collection \(\{GU_i : i\leq m\}\) covers \(X\) we have that \(\bigcap_{i=1}^m V_iU_i\not= \emptyset\). In this paper, the authors prove among other things the following result. If \(X\) is a paracompact \(G\)-space with \(\text{gen}(X,G)=k\), then the collection \(\{V_i : i\leq k\}\) with \(V_1=\{e\}\) and \(V_i = G^*\) for \(i\geq 2\) is an intersection family for \(X\). The authors also discuss how this theorem might be improved, and illustrate their results by a free action of the Klein four-group on the torus.
scientific article; zbMATH DE number 1879123
scientific article