About non-uniqueness when removing closed orbits in Morse-Smale vector fields

TL;DR

This paper investigates the non-uniqueness in the process of modifying Morse-Smale vector fields by removing closed orbits, revealing multiple ways to perform such modifications and discussing implications for topological and algebraic structures.

Contribution

It demonstrates the existence of multiple non-equivalent procedures for removing closed orbits in Morse-Smale vector fields and explores their impact on canonical topological and algebraic assignments.

Findings

Multiple non-equivalent methods for orbit removal exist.

Non-uniqueness affects the assignment of CW and chain complexes.

Illustrative examples demonstrate the non-uniqueness phenomenon.

Abstract

Given a Morse-Smale vector field on a smooth manifold, Franks described how one can replace a closed orbit of index $k$ by two rest points of index $k+1$ and $k$, using a local perturbation. Combined with classical results about gradient-like vector fields, this gives a method of assigning different topological or algebraic structures to Morse-Smale vector fields. We show that there are multiple non-equivalent ways of following this procedure and illustrate this non-uniqueness in various examples. We describe the consequences of this non-uniqueness to the endeavour of assigning CW complexes or chain complexes to Morse-Smale vector fields in a canonical way.