Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category, or simply, category) of a topological space X is the topological invariant defined to be the smallest cardinality of an index set I such that there is an open covering \{U_i\}_{i\in I} of X with the property that for each i\in I, the inclusion map U_i\hookrightarrow X is nullhomotopic. For example, if X is the circle, this takes the value two.

Recently a different normalization of the invariant has been adopted, which is one less than the original definition by Lusternik and Schnirelmann. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).

In general it is not so 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 LS category.

It was, as originally defined for the case of X a manifold, the lower bound for the number of critical points that a real-valued function on X 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).

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