Transverse tori in Engel manifolds

TL;DR

This paper explores the behavior and classification of transverse tori in Engel 4-manifolds, revealing their rich invariants and demonstrating their abundance and complexity similar to knots in contact 3-manifolds.

Contribution

It introduces a classification of formal invariants for transverse tori in Engel manifolds and shows their completeness as obstructions in overtwisted structures, expanding understanding of transverse submanifolds.

Findings

Every trivial normal bundle torus is isotopic to infinitely many transverse tori.

Formal invariants are richer than for transverse knots and classify these tori.

Many Engel manifolds have infinitely many transverse homotopy and isotopy classes.

Abstract

We show that tori in Engel 4-manifolds behave analogously to knots in contact 3-manifolds: Every torus with trivial normal bundle is isotopic to infinitely many distinct transverse tori, distinguished locally (and globally in the nullhomologous case) by their formal invariants. (Few examples of transverse tori were previously known.) We classify the formal invariants, which are richer than for transverse knots. We show that in an overtwisted Engel structure, up to homotopy through such structures, these invariants are a complete set of uniqueness obstructions, and every torus with trivial normal bundle can be made transverse realizing any combination of these invariants. Fixing Engel structures not known to be overtwisted, we explore the range of the primary invariants of given tori. A sample application is that many Engel manifolds admit infinitely many transverse homotopy classes of unknotted transverse tori such that each class contains infinitely many transverse isotopy classes.