Topology change of levels sets in Morse theory

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...

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Bibliographic Details
Published in:arXiv.org 2019-10
Main Authors: Knauf, Andreas, Martynchuk, Nikolay
Format: Article
Language:English
Subjects:
Celestial mechanics
Critical point
Manifolds (mathematics)
Topology
Online Access:Get full text
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Summary: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.
ISSN:2331-8422