Volume 25, issue 2 (2021)

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Mayer–Vietoris property for relative symplectic cohomology

Umut Varolgunes

Geometry & Topology 25 (2021) 547–642

We construct a Hamiltonian Floer theory-based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of restriction maps, and prove some basic properties. Our main contribution is to identify natural geometric conditions in which relative symplectic cohomology of two subsets satisfies the Mayer–Vietoris property. These conditions involve certain integrability assumptions involving geometric objects called barriers — roughly, a 1–parameter family of rank 2 coisotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (ie constant) Floer solutions are the main actors.

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Floer theory, involutive systems, descent
Mathematical Subject Classification 2010
Primary: 53D40
Secondary: 53D37, 81S10
Received: 2 June 2018
Revised: 21 February 2020
Accepted: 13 May 2020
Published: 27 April 2021
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Mark Gross
Umut Varolgunes
Department of Mathematics
Stanford University
Stanford, CA
United States