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arXiv · 1910.05294

Topology change of levels sets in Morse theory

Andreas Knauf·Nikolay Martynchuk
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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.

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BibTeXRIS

Andreas Knauf, Nikolay Martynchuk. 2019-10-11. Topology change of levels sets in Morse theory. https://arxiv.org/abs/1910.05294

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