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 $A_{\infty }$–structure on the underlying wrapped Floer complex, and (under suitable assumptions) an $A_{\infty }$–homomorphism realizing the restriction to a Liouville subdomain. The construction of the $A_{\infty }$–structure relies on an implementation of homotopy direct limits, and involves some new moduli spaces which are solutions of generalized continuation map equations.