A quick proof of Singhof's cat(M �S1)=cat(M)+1 theorem

Abstract

For a topological space X, the Lusternik-Schnirelmann 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. W. Singhof [6] proved that the minimal number of n-balls which suffice to cover a closed PL n-manifold M coincides with the Lusternik-Schnirelmann 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×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.