A Note on Somewhere Positive Loops of Contactomorphisms

TL;DR

This paper explores the properties of certain contractible loops of contactomorphisms in contact geometry, focusing on the significance of immaterial subsets and their complements.

Contribution

It introduces the concept of immaterial subsets and analyzes the contact geometric implications of their complements, supported by results on symplectic homology and contact quasi-measures.

Findings

Complement of Reeb-invariant immaterial subset is large in contact geometric terms.

Symplectic homology of the filling relates to these subsets.

Contact quasi-measures provide insights into the structure of contactomorphism loops.

Abstract

In this note, we consider contractible loops of contactomorphisms that are positive over some non-empty closed subset of a contact manifold. Such closed subsets are called immaterial. We argue that the complement of a Reeb-invariant immaterial subset can be seen as big in contact geometric terms. This is supported by two results: one regarding symplectic homology of the filling and the other regarding recently introduced contact quasi-measures.