Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

Abstract

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be non-exact and non-compactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.

The symplectic cohomology of magnetic cotangent bundles

“…For the relevant definitions see Section 3 below. Remark: During the course of the preparation of this article, Benedetti and Ritter released [11], which proves similar results to ours in the case of surfaces.…”

“…This would be slightly more natural in light of the fact that our symplectic cohomology groups SH * (T * N; ω σ ) depend on σ only through the half-line {r 11). Nevertheless, we stick with polytopes to be consistent with [46].…”

“…where the first map is induced by the evaluation LN ×S 1 → N, (q, t) → q(t), and the second projects onto the Künneth summand. Explicitly, if k ∈ H 2 (N; R) and σ is a closed 2-form with [σ] = k, then a closed 1-form τ (σ) ∈ Ω 1 (LN) representing the cohomology class τ (k) is given by (11) τ (σ) q (v) :=…”

“…This uses properties of the BV operator on symplectic cohomology. See for instance [11] for details.…”