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 homotopical invariant defined to be the smallest integer number $k$ such that there is an open covering $\{U_{i}\}_{1\leq i\leq k}$ of $X$ with the property that each inclusion map $U_{i}\hookrightarrow X$ is nullhomotopic. For example, if $X$ is the circle, 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 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).

See also

• Ganea conjecture
• Systolic category

References

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

• Samuel Eilenberg, Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups, Annals of Mathematics, 2nd Ser., 65 (1957), no. 3, 517 – 518

• 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.

• Tudor Ganea, Some problems on numerical homotopy invariants, Lecture Notes in Math. 249 (Springer, Berlin, 1971), pp. 13 – 22 MR0339147

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

• Kathryn Hess, A proof of Ganea's conjecture for rational spaces, Topology 30 (1991), no. 2, 205--214. MR1098914

• Norio Iwase, "Ganea's conjecture on Lusternik-Schnirelmann category", in Bulletin of the London Mathematical Society, 30 (1998), no.6, 623 – 634 MR1642747

• Norio Iwase, A_∞-method in Lusternik-Schnirelmann category, Topology 41 (2002), no. 4, 695--723. MR1905835

• Lucile Vandembroucq, Fibrewise suspension and Lusternik-Schnirelmann category, Topology 41 (2002), no. 6, 1239--1258. MR1923222

• Octav Cornea, Gregory Lupton, John Oprea, Daniel Tanré, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs, 103. American Mathematical Society, Providence, RI, 2003 ISBN 0-8218-3404-5