Equivariant Floer cohomology for contactomorphisms of quotient spaces

TL;DR

This paper introduces an equivariant contact Floer cohomology framework to prove the orderability of certain quotient contact manifolds, extending Givental's nonlinear Maslov index.

Contribution

It develops an equivariant Floer cohomology theory and an analogue of Givental's index for quotient contact manifolds, advancing the understanding of their orderability.

Findings

Proves orderability of contact manifolds as quotients of fillable manifolds.

Develops an equivariant contact Floer cohomology theory.

Constructs an analogue of Givental's nonlinear Maslov index.

Abstract

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.