Equivariant Floer cohomology for contactomorphisms of quotient spaces

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.

Figures (17)

• Figure 1: Elements of $\mathscr{M}(\Delta^{2})$.
• Figure 2: Cubical coordinates on the moduli space of straight line paths are ultimately derived from the points $p_{1}$ and $p_{2}$.
• Figure 3: Graph of auxiliary functions $f:\mathbb{R}\to \mathbb{R}$ and $\rho:\mathbb{R}\to \mathbb{R}$. We require the derivative $\rho'(t)$ to be supported in the interior of the interval where $f=1$.
• Figure 4: Illustration of a solution to \ref{['eq:zero-simplex-eqn-mod-2']}. With our cohomological conventions, $\gamma_{+}$ will be considered as the "input."
• Figure 5: Illustration of the domain of a solution in $\mathscr{M}(\Phi,k,\pm)$, when $\Phi$ is a $n$-simplex. The equation a solution $(\ell,\pi,u)$ must solve is expressed in terms of $n$ sub-intervals $[s_{i},s_{i}+w_{i}]$ determined by $\ell$, and the values of $\Phi(\ell)$.

Theorems & Definitions (129)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Proposition 1.4
• Proposition 1.5
• proof : Proof of Proposition \ref{['proposition:lipschitz']}
• proof : Proof of Proposition \ref{['proposition:orderable']}
• Remark 1.6
• Remark 1.7: Hypertightness
• Remark 1.8: Reeb orbits