Extended Fuller index, sky catastrophes and the Seifert conjecture

Abstract

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] 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 [Formula: see text]. 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.