A quick proof of Singhof's \(cat(M\times S^ 1)=cat(M)+1\) theorem (Q1099841)

For a topological space X, the Lyusternik-Shnirel'man category, cat(X), is the smallest number N such that X can be covered by N open subsets each of which is contractible in X. \textit{W. Singhof} [ibid. 29, 385-415 (1979; Zbl 0415.55001)] proved that the minimal number of n-balls which suffice to cover a closed PL n-manifold M coincides with the Lyusternik- Shnirel'man category if the latter is not too small compared with the dimension of M. As a consequence, one obtains in this case that \(cat(M\times S^ 1)=cat(M)+1\), thus establishing a special case of a long-standing conjecture. The purpose of this paper is to give a quick proof of Singhof's results by exploiting the linear structure between the k-skeleton of a polyhedron and its dual skeleton.