Foliated vector fields without periodic orbits

TL;DR

This paper explores the relationship between vector fields tangent to foliations and the absence of closed orbits, establishing a homotopy equivalence in higher-dimensional leaves and contrasting with surface foliations in 3-manifolds.

Contribution

It introduces parametric versions of Wilson's and Kuperberg's plugs and proves a homotopy equivalence between spaces of vector fields with and without closed orbits for high-dimensional foliations.

Findings

Homotopy equivalence for vector fields on foliations with leaves of dimension ≥ 3

Contrast with foliations by surfaces in 3-manifolds

Extension of Wilson's and Kuperberg's plug techniques

Abstract

In this article parametric versions of Wilson's plug and Kuperberg's plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and the subspace of those without closed orbits, as long as the leaves of the foliation have dimension at least 3. We contrast this with the case of foliations by surfaces in 3-manifolds.