A note on Hempel--McMillan coverings of 3-manifolds
The paper provides a new proof, as well as a slight generalization, of a result of [\textit{J. P. Hempel} and \textit{D. R. McMillan jun.}, Fundam. Math. 64, 99--104 (1969; Zbl 0176.53302)], saying that a closed \(3\)-manifold which is a union of three open \(n\)-punctured balls (\(n\geq 0\)) must be a connected sum of \(S^3\) and \(S^2\)-bundles over \(S^1\). The facts are reinterpreted from the point of view of the notion of \(A\)-category due to [\textit{M. Clapp} and \textit{D. Puppe}, Trans. Am. Math. Soc. 298, 603--620 (1986; Zbl 0618.55003)]. When \(A\) is a point, this is just the classical Lusternik-Shnirelmann category. The authors are concerned here with the case that \(A\) is a point or the \(2\)-sphere \(S^2\).
coverings of \(3\)-manifolds by open subsets
scientific article; zbMATH DE number 5144523