Infinite Dimensional Morse Theory

The basic results in Morse theory are the Morse inequalities and the Morse handle body theorem. They are established on the Banach Finsler manifolds or on the Hilbert Riemannian manifolds in Section 4. The tool in this study is the deformation theorem, which is introduced in Section 3. Some preliminaries on algebraic topology and on infinite dimensional manifolds are reviewed in Sections 1 and 2 respectively. Readers who are familiar with the background material may skip over these two sections. Gromoll-Meyer theory on isolated critical points plays an important role in the applications of Morse theory because the nondegeneracy assumption in the handle body theorem might not hold for concrete problems. Section 5 is devoted to introducing Gromoll-Meyer theory systematically and examines the splitting lemma, the homotopy invariance theorem, the shifting theorem, and the Marino Prodi approximation theorem. The rest of the chapter consists of the extensions of the basic results of Morse theory in different directions: in Section 6.1, to the extension to manifolds with boundaries as well as to the functions satisfying certain boundary value conditions, in Section 6.2, to the extension from manifolds to the locally convex closed subsets; and, in Section 7, to functions with symmetry under a compact Lie group action. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.