Topology change of levels sets in Morse theory

TL;DR

This paper investigates how the topology of level sets in Morse theory changes at critical points, providing conditions and examples, especially for Hamiltonian functions on cotangent bundles.

Contribution

It establishes new conditions for topology change in level sets, including special cases for Hamiltonian functions, and discusses applications and counterexamples.

Findings

Topology of regular level sets changes at critical points unless the index is half the dimension.

For Hamiltonian functions, topology change relates to the configuration space topology.

Counterexamples and applications to celestial mechanics are provided.

Abstract

Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty, a]$ of a Morse function $f: M \to R$, 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 function, 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 configuration space. (Counter-)examples and applications to celestial mechanics are also discussed.