Topology change of level sets in Morse theory

Abstract

The classical Morse theory proceeds by considering sublevel sets $f^{-1} (-\infty, a]$ of a Morse function $f : M \to \mathbb{R}$, 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.