A note on somewhere positive loops of contactomorphisms

TL;DR

This work studies immaterial subsets in contact geometry and shows that their complements are 'big' in a contact-geometric sense. It blends selective symplectic homology with recent contact quasi-measures to establish surjectivity of continuation maps for complements of immaterial, Reeb-invariant sets and to derive lower bounds on quasi-measures via spectral invariants. A key outcome is that immaterial preimages under Reeb-invariant maps force the complement to be non-displaceable, yielding concrete examples in cotangent unit bundles. The results extend previous selective SH work and provide practical criteria for non-displaceability, with implications for orderability and rigidity phenomena in contact manifolds.

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.

Theorems & Definitions (5)

• proof
• proof
• proof : Proof of Theorem \ref{['thm:complement-big']}
• proof
• proof : Proof of Theorem \ref{['thm:tau']}