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Topology change of level sets in Morse theory

Andreas Knauf Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany Nikolay Martynchuk Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany; and Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

Dynamical Systems Algebraic Topology and General Topology mathscidoc:2203.11003

Arkiv for Matematik, 58, (2), 333-356, 2020.11
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.
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@inproceedings{andreas2020topology,
  title={Topology change of level sets in Morse theory},
  author={Andreas Knauf, and Nikolay Martynchuk},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311100752085888946},
  booktitle={Arkiv for Matematik},
  volume={58},
  number={2},
  pages={333-356},
  year={2020},
}
Andreas Knauf, and Nikolay Martynchuk. Topology change of level sets in Morse theory. 2020. Vol. 58. In Arkiv for Matematik. pp.333-356. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311100752085888946.
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