Extensible positive loops and vanishing of symplectic cohomology

Abstract

The symplectic cohomology of certain symplectic manifolds $W$ with non-compact ends modelled on the positive symplectization of a compact contact manifold $Y$ is shown to vanish whenever there is a positive loop of contactomorphisms of $Y$ which extends to a loop of Hamiltonian diffeomorphisms of $W$. An open string version of this result is also proved: the wrapped Floer cohomology of a Lagrangian $L$ with ideal Legendrian boundary $Λ$ is shown to vanish if there is a positive loop $Λ_{t}$ based at $Λ$ which extends to an exact loop of Lagrangians based at $L$. Various examples of such loops are considered. Applications include the construction of exotic compactly supported symplectomorphisms and exotic fillings of $Λ$.

Figures (6)

• Figure 1: Differential is defined by counting solutions to Floer's equation on the cylinder.
• Figure 2: A morphism between two contact-at-infinity systems is a homotopy class of extensions of $\psi_{0,t},\psi_{1,t}$ to $\psi_{s,t}$. The restriction to the top $t=1$ is required to be non-negative with respect to $s$.
• Figure 3: Figure used in proof of (i) with $k=5$.
• Figure 4: The moduli spaces used to prove Theorem \ref{['theorem:pss-factor']}. These induce morphism $\mathrm{HF}(R_{-\epsilon t})\to \mathrm{HF}(R_{\epsilon t})$ by considering the positive puncture as the input.
• Figure 5: More detailed view of the third piece of Figure \ref{['fig:PSS']}.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Lemma 1.6
• Theorem 1.7
• Conjecture 1.8
• Lemma 1.9
• Lemma 1.10