Categoría de lusternik-schnirelmann y funciones de morse en los espacios simétricos

Abstract

The initial purpose of this dissertation was to compute the LS category of several Lie groups and symmetric spaces. One of the methods to study this topological invariant is the one used by W. Singhof in 1975 to compute the category of the unitary groups. Singhof obtains an explicit covering by n categorical open sets related with the eigenvalues. When we try to extend Singhof¿s method to the symplectic group Sp(n) it becomes clear that the needed condition is related with left eigenvalues. To prove that those open sets are contractible I use the Cayley transform. It should be kept in mind that in a compact manifold the LS category (plus one) is a lower bound for the number of critical points of any smooth function. This fact compelled me to study the Morse-Bott functions and to give local charts for its critical manifold by means of the Cayley transform. Using this results, I obtain explicit contractible coverings for U(n)/O(n) and U(2n)/Sp(n) as well as a formula that relates catSp(n) with the critical submanifolds of the height function.