An open string analogue of Viterbo functoriality

Abstract

Liouville domains are a special type of symplectic manifolds with boundary (they have an everywhere defined Liouville flow, pointing outwards along the boundary). Symplectic cohomology for Liouville domains was introduced by Cieliebak-Floer- Hofer-Wysocki and Viterbo. The latter constructed a restriction (or transfer) map associated to an embedding of one Liouville domain into another. In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called "wrapped Floer cohomology". We construct an A∞-structure on the underlying wrapped Floer complex, and (under suitable assumptions) an A∞- homomorphism realizing the restriction to a Liouville subdomain. The construction of the A∞-structure relies on an implementation of homotopy direct limits, and involves some new moduli spaces which are solutions of generalized continuation map equations.