Relative H-Principle and Contact Geometry

TL;DR

This paper explores the relative h-principle in contact geometry, showing that under certain conditions, the space of solutions admits a homotopy section, and applies this to find infinite cyclic subgroups in contactomorphism groups.

Contribution

It establishes a homotopy section result for spaces satisfying a relative h-principle and applies it to overtwisted contact structures to identify infinite cyclic subgroups.

Findings

Homotopy section exists for certain solution spaces under the relative h-principle.

Application to overtwisted contact structures yields infinite cyclic subgroups.

Advances understanding of the topology of contactomorphism groups.

Abstract

We show that if $F(M)$ is some space of holonomic solutions with space of formal solutions $F^f(M)$ that satisfies a certain relative $h$-principle, then the non-relative map $F(M) \to F^f(M)$ admits a section up to homotopy. We apply this to the relative $h$-principle for overtwisted contact structures proved by Borman-Eliashberg-Murphy to find infinite cyclic subgroups in the homotopy groups of the contactomorphism group of $M$.