Remarks on eternal classes in symplectic cohomology

TL;DR

This paper investigates special eternal classes in symplectic cohomology, their properties, and implications for contactomorphism groups, providing criteria for their existence and non-existence in certain symplectic manifolds.

Contribution

It introduces the concept of eternal classes in symplectic cohomology, analyzes their properties, and establishes criteria for their existence or absence in specific symplectic manifolds.

Findings

Eternal classes are in the image of all continuation maps and never die.

Spectral invariants of non-eternal classes are sub-additive under the pair-of-pants product.

Criteria for existence include the presence of certain Lagrangians; criteria for non-existence include displaceability of compact sets.

Abstract

This paper studies special classes in the symplectic cohomology of a semipositive and convex-at-infinity symplectic manifold $W$. The classes under consideration lie in the image of every continuation map (for this reason, we call them eternal classes as they are never born and never die). Non-eternal classes in symplectic cohomology can be used to define spectral invariants for contact isotopies of the ideal boundary $Y$ of $W$. It is shown that the spectral invariants of non-eternal classes behave sub-additively with respect to the pair-of-pants product. This is used to define a spectral pseudo-metric on the universal cover of the group of contactomorphisms. We also give criteria for existence and non-existence of eternal classes. First, a compact monotone Lagrangian with odd Euler characteristic and minimal Maslov number at least $2$ implies the existence of non-zero eternal classes (e.g., $T^{*}\mathrm{RP}^{2n}$ has non-zero eternal classes). Second, no non-zero eternal classes exist if every compact set in $W$ is smoothly displaceable (e.g., $T^{*}T^{n}$ has no non-zero eternal classes).