Topology change of level sets in Morse theory - Knauf, Andreas; Martynchuk, Nikolay (Arkiv For Matematik, 2020)
The 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 the topology of f−1(a) 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 f−1(a) 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 base space. (Counter-)examples and applications to celestial mechanics are also discussed.
CRIS-Team Scopus Importer:2020-11-20
Keywords: Hamiltonian and celestial mechanics; invariant manifolds; Morse theory; Surgery theory; Vector bundles