Appendix C - Category and genus

Summary

Every human activity, good or bad, except mathematics, must come to an end.

Paul Erdös (1913–1996)

Topological tools play a central role in the study of variational problems. Though this approach was foreshadowed in the works of Poincaré and Birkhoff, the force of these ideas was realized in the first decades of the twentieth century, in the pioneering works of Ljusternik and Schnirelmann [145] and Morse [163, 164]. In this section we recall the notions of Ljusternik–Schnirelmann category and Krasnoselski genus as well as some of their basic properties.

Definition C.1. Let M be a topological space and let A ⊂ M be a subset. The continuous map η: A × [0, 1] → M is called a deformation of A in M if η(u, 0) = u for every u ∈ A. The set A is said be contractible in M if there exists a deformation η: A × [0, 1] → M with η(A, 1) = {p} for some p ∈ M.

Definition C.2. Let M be a topological space. A set A ⊂ M is said to be of Ljusternik–Schnirelmann category k in M (denoted cat_M(A) = k) if it can be covered by k but not by k − 1 closed sets which are contractible to a point in M. If such a k does not exist, then cat_M(A) = ∞.