Morse-Theorie für Niveaumengen
arXiv-Artikel mit Nikolay Martynchuk
Classical Morse theory proceeds by considering sublevel sets f -1((-∞, a]) of a Morse function f: M → ℝ, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f -1(a) and give conditions under which their topology changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M.
When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the configuration space. (Counter-)examples and applications to celestial mechanics are also discussed.