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arXiv · 1703.07801

Extended Fuller index, sky catastrophes and the Seifert conjecture

Yasha Savelyev
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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.

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Yasha Savelyev. 2019-01-27. Extended Fuller index, sky catastrophes and the Seifert conjecture. https://arxiv.org/abs/1703.07801

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