Lusternik–Schnirelmann category
In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space formula_1 is the homotopy invariant defined to be the smallest integer number formula_2 such that there is an open covering formula_3 of formula_1 with the property that each inclusion map formula_5 is nullhomotopic. For example, if formula_1 is a sphere, this takes the value two. Sometimes a different normalization of the invariant is adopted, which is one less than the definition above. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below). In  general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for the LS-category. It was, as originally defined for the case of formula_1 a manifold, the lower bound for the number of critical points that a real-valued function on formula_1 could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers  is a lower bound for the number of critical points of a Morse function). The invariant has been generalized in several different directions (group actions, foliations, simplicial complexes, etc.).