Topology change of level sets in Morse theory

Knauf A, Martynchuk N (2020)

Publication Type: Journal article

Publication year: 2020

Journal

Arkiv For Matematik Royal Swedish Academy of Sciences, Institut Mittag-Leffler

Book Volume: 58

Pages Range: 333-356

Journal Issue: 2

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

Abstract

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.

Authors with CRIS profile

Andreas Knauf Lehrstuhl für Mathematik (Mathematische Physik) Nikolay Martynchuk Lehrstuhl für Mathematik (Mathematische Physik)

How to cite

APA:

Knauf, A., & Martynchuk, N. (2020). Topology change of level sets in Morse theory. Arkiv For Matematik, 58(2), 333-356. https://dx.doi.org/10.4310/ARKIV.2020.v58.n2.a6

MLA:

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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