Topology change of level sets in Morse theory

Knauf A, Martynchuk N (2020)

Publication Type: Journal article

Publication year: 2020


Book Volume: 58

Pages Range: 333-356

Journal Issue: 2

DOI: 10.4310/ARKIV.2020.v58.n2.a6


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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Knauf, A., & Martynchuk, N. (2020). Topology change of level sets in Morse theory. Arkiv For Matematik, 58(2), 333-356.


Knauf, Andreas, and Nikolay Martynchuk. "Topology change of level sets in Morse theory." Arkiv For Matematik 58.2 (2020): 333-356.

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