# Extended Fuller index, sky catastrophes and the Seifert conjecture

## TL;DR

This paper extends the Fuller index to a broader class of vector fields on compact manifolds, proving the existence of periodic orbits under certain homotopies without sky catastrophes, and constrains sky catastrophes in Reeb vector fields.

## Contribution

It generalizes the Fuller index and establishes conditions ensuring periodic orbits during homotopies, addressing a longstanding question in dynamical systems.

## Key findings

- Extended Fuller index applicable to new class of vector fields.
- Proved existence of periodic orbits under homotopies without sky catastrophes.
- Constrained sky catastrophes in Reeb vector fields.

## Abstract

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields $X$ on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at $X$ has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on $S ^{2k+1} $. A sky catastrophe, is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller's foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.