Arkiv för Matematik

Volume 58 (2020)

Number 2

Topology change of level sets in Morse theory

Pages: 333 – 356

DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n2.a6

Authors

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)

Abstract

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.

Keywords

invariant manifolds, Hamiltonian and celestial mechanics, Morse theory, surgery theory, vector bundles

2010 Mathematics Subject Classification

37N05, 55R25, 57N65, 57R65, 58E05, 70F10, 70H33

Received 11 March 2020

Received revised 16 July 2020

Accepted 29 July 2020