Finitude homotopique et isotopique des structures de contact tendues

TL;DR

This paper proves that on closed 3-manifolds, the homotopy classes of plane fields containing tight contact structures are finite, and if the manifold is atoroidal, the isotopy classes are also finite.

Contribution

It establishes finiteness results for homotopy and isotopy classes of tight contact structures on closed 3-manifolds, extending understanding of contact topology.

Findings

Finite homotopy classes of tight contact structures on closed 3-manifolds.

Finite isotopy classes of tight contact structures on atoroidal 3-manifolds.

Advances the classification of contact structures in 3-dimensional topology.

Abstract

Let V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number.