Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category of a topological space X is the topological invariant defined as the smallest cardinality of an open covering of X by contractible subsets. For example, if X is the circle, this takes the value two.

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. It was, as originally defined for the case of X a manifold, the lower bound for the number of critical points a Morse function on X could possess (cf. Morse theory).

References

• [1] R. H. Fox, On the Lusternik-Schnirelmann category, Annals of Mathematics 42 (1941), 333-370.

• I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331-348.

• [2] F. Takens, The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category, Inventiones Mathematicae 6 (1968), 197-244.