Mayer–Vietoris property for relative symplectic cohomology

Abstract

We formulate and prove a chain level descent property of symplectic cohomology for involutive covers by compact subsets that take into account the natural algebraic structures that are present. The notion of an involutive cover is reviewed. We indicate the role that the statement plays in mirror symmetry.

Super-rigidity of certain skeleta using relative symplectic cohomology

“…The first two statements are straightforward. For the third statement, note that we can choose an acceleration data for K, so that all the 1-chords of L lie outside K, and moreover the value of the Hamiltonian H n at the chords of H n is approximately n. Now by the "adiabatic" argument in [17], we obtain a uniform lower bound on the topological energies of all possible continuation maps for slowed down acceleration data, which proves that none of the generators survive in the completion. For the last one, note that the proof from [18] applies verbatim here as one simply replace the 1-periodic orbits in Lemma 4.1.1 from [18] with 1-chords on L, and the acceleration data that is constructed there would also satisfy this modified requirement.…”

“…So Φ is a chain isomorphism. To show (17), in view of (18) it remains to show that there exists a constant C such that:…”

“…We use H = (H 1 i , H 0 i ) for the vertical arrows, and H = (H 1 i , H 0 i+1 ) for the diagonal arrows. (Recall that we use 1-dimensional families of data for the latter, as usual [17]. )…”

“…The isomorphism is explicitly constructed using what we call the contact Fukaya trick. Using the argument in Lemma 4.1.1 of [17] we can actually show that this is isomorphism is given by the restriction map as we had initially asked for. Proving this would make Section 4 even more technical and this strengthening does not help us in the paper.…”

“…The space S c contains the usual continuation map data coming from monotone homotopies: namely, α = dt and H ≥ 0 is non-decreasing in s.5.2. Extrinsic continuation.We assume that the reader is familiar with the algebra from[17]; also see Section 3.…”