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October 2020 Topology change of level sets in Morse theory
Andreas Knauf, Nikolay Martynchuk
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Ark. Mat. 58(2): 333-356 (October 2020). DOI: 10.4310/ARKIV.2020.v58.n2.a6


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.


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Andreas Knauf. Nikolay Martynchuk. "Topology change of level sets in Morse theory." Ark. Mat. 58 (2) 333 - 356, October 2020.


Received: 11 March 2020; Revised: 16 July 2020; Published: October 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ARKIV.2020.v58.n2.a6


Rights: Copyright © 2020 Institut Mittag-Leffler


Vol.58 • No. 2 • October 2020
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