02-2026
This paper establishes the orderability of contact manifolds which are quotients of fillable contact manifolds under finite group actions compatible with the filling, the prototypical example being $\mathbb{R}P^{2n-1}$ as the quotient of $S^{2n-1}$. Our approach employs an equivariant formulation of the so-called contact Floer cohomology theory. This leads us to develop an analogue of Givental's nonlinear Maslov index using the $\mathbf{k}[[x]]$-module structure on an equivariant version of contact Floer cohomology. A key idea is that mapping cones of continuation maps detect crossings with the discriminant (recall that Givental's index is a continuous integer valued function on the complement of the discriminant). To properly handle the inherent non-canonicity in defining such mapping cones, we lift the structure of contact Floer cohomology to chain level by defining it as an $\infty$-functor on a suitable $\infty$-categorification of the Eliashberg-Polterovich orderability relation on the universal cover of the contactomorphism group.
@misc{inftyeq,
author = {Dylan Cant and Eric Kilgore and Jun Zhang},
title = {Equivariant {Floer} cohomology for contactomorphisms of quotient spaces},
year = {2026},
eprint = {2602.21152},
archivePrefix = {arXiv},
primaryClass = {math.SG},
}