Category (in the sense of Lyusternik-Shnirel'man)

Lyusternik–Shnirel'man category

A characteristic of a topological space $E$: the minimal number $\mathrm{cat}\,E$ of closed sets $A_i \subset E$ covering $E$, each of which can be contracted to a point by means of a continuous deformation in $E$. Lyusternik–Shnirel'man category is a homotopy invariant (that is, it is the same for all topological spaces of the same homotopy type). It is of special importance for variational calculus in the large, since it provides a lower bound for the number of stationary (critical) points of a smooth function on a closed manifold. Even the simplest examples, on non-closed manifolds, which may happen to have no critical points, such as the function $f(x)=x$ considered on the whole real axis, show that outside the class of closed manifolds this estimate can no longer be expected to be valid for all smooth functions. Nevertheless, in a number of cases it is possible to obtain a similar bound for the number of critical points of various functionals studied in the calculus of variations, considered as functions on an infinite-dimensional function space. There are no general methods for calculating the Lyusternik–Shnirel'man category (although its value is known for certain concrete spaces); in general only the following estimate is known: $$ \mathrm{cat}\,E \ge \mathrm{long}\,E + 1 \,, $$ where the (cohomological) length $\mathrm{long}\,E$ is defined as the largest number of cohomology classes of positive dimension whose product is non-zero. Therefore the arguments are sometimes carried out directly in terms of the cohomological product (or, dually, in terms of intersection of cycles) without referring to the category.

Category was introduced in [1]. It was calculated for the first time in a non-trivial case (for a real projective space) by L.G. Shnirel'man, after which it was immediately used for the solution of a number of problems, among them, the problem of Poincaré on three closed geodesics on surfaces homeomorphic to a $2$-sphere [2].

References

[1]	L. [L.A. Lyusternik] Lusternik, "Sur quelques méthodes topologique dans la géométrie différentielle" , Atti congress. internaz. mathematici (Bologna, 1928) , 4 , Zanichelli (1931) pp. 291–296
[2]	L.A. Lyusternik, L.G. Shnirel'man, "Topological methods in variational problems and their application to the differential geometry of surfaces" Uspekhi Mat. Nauk , 2 : 1 (1947) pp. 166–217 (In Russian)

Comments

The Lyusternik–Shnirel'man theorem on closed geodesics states that on the $2$-dimensional sphere with an arbitrary Riemannian metric there exist three closed geodesics without self-intersections [a2], p. 214.

References

[a1]	L. [L.A. Lyusternik] Lusternik, L. [L.G. Shnirel'man] Schnirelman, "Méthodes topologiques dans les problèmes variationels" , Hermann (1934)
[a2]	W. Klingenberg, "Lectures on closed geodesics" , Springer (1978)
[a3]	H. Seifert, W. Threlfall, "Variationsrechnung im Groszen (Morsesche Theorie)" , Chelsea, reprint (1948) pp. 91–92