Lusternik-Schnirelmann category (or LS-category) of a space \(X\) is defined as the least number \(n\) such that there is a covering of \(X\) by \(n+1\) open subsets each of which can be retracted to a point inside \(X\). This invariant is a lower bound for the number of critical points of any smooth function on a manifold. If the property “contractible in \(X\)” is replaced by “contractible”, we have to take the minimum on all spaces with the same homotopy type as \(X\) for having a homotopical invariant. In this case, we still have an invariant very close to the LS-category since the difference between them is at most 1. The new invariant introduced in this work seems related to the previous one but it is deeply different.
First, recall that a good cover of a space is a family of open subsets such that every non-empty intersection is contractible. Then the covering type of \(X\) is defined as the minimum number of elements in a good cover of a space \(X'\) homotopic to \(X\). By definition, this is a homotopical invariant. The difference between LS-category and covering type can be as large as we want since a sphere \(S^n\) has covering type equal to \(n+2\) and LS-category equal to 1.
In the paper under review, the authors provide a definition equivalent to closed cover and study the covering type of some examples. They give explicit values for the torus and for the real projective space. For a more general surface, they give a bound on it as a function of the genus. They also list some open questions on this subject.
Note also the existence of a recent paper on this invariant by D. Govc et al. [“Estimates of covering type and the number of vertices of minimal triangulations”, Preprint, arXiv:1710.03333].