Strong orderability or assumption* for standard contact lens spaces

Determine whether the standard contact lens spaces L_k^{2n+1}(\underline{w}) endowed with the standard contact structure \xi_k^{\underline{w}} are, in general, strongly orderable or satisfy assumption*, where assumption* means that the contact manifold admits a strong symplectic filling by a weakly^+-monotone symplectic manifold whose unit in symplectic homology is not eternal. Establishing either property would clarify whether the contact big fiber result for lens spaces can be deduced from existing frameworks for strongly orderable manifolds or from results assuming assumption*.

Background

The paper proves a contact big fiber theorem for strongly orderable closed contact manifolds and, separately, for standard contact lens spaces using spectral selectors arising from generating functions and Givental’s nonlinear Maslov index. The authors emphasize that existing approaches to big fiber theorems often rely on symplectic homology techniques encoded in assumption*, while their construction for lens spaces uses different tools.

Whether lens spaces fall into either of the two broad classes—strongly orderable or satisfying assumption*—is central for relating the lens space big fiber theorem to earlier general frameworks. Clarifying this would determine if the lens space result can be deduced from (rather than proved independently of) those frameworks.