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# 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

#### 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

Published 3 November 2020