Spectral selectors on strongly orderable contact manifolds and applications

Abstract

We prove that the spectral selectors introduced by the author for closed strongly orderable contact manifolds satisfy algebraic properties analogous to those of the spectral selectors for lens spaces constructed by Allais, Sandon and the author using Givental's nonlinear Maslov index. As applications, first we establish a contact big fiber theorem for closed strongly orderable contact manifolds as well as for lens spaces. Second, when the Reeb flow is periodic, we construct a stably unbounded conjugation invariant norm on the contactomorphism group universal cover. Moreover, when all its orbits have the same period, we show that the Reeb flow is a geodesic for the discriminant and oscillation norms of Colin-Sandon. Finally, we generalize a theorem of Albers-Fuchs-Merry stating that non-orderability implies the Weinstein conjecture.