An open string analogue of Viterbo functoriality

TL;DR

This paper develops an analogue of Viterbo functoriality for wrapped Floer cohomology of Lagrangian submanifolds with Legendrian boundary in Liouville domains, introducing new A_infinity-structures and homomorphisms.

Contribution

It constructs an A_infinity-structure on wrapped Floer complexes and defines restriction maps for Lagrangian submanifolds, extending symplectic cohomology functoriality to a new setting.

Findings

Constructed an A_infinity-structure on wrapped Floer complexes.

Defined A_infinity-homomorphisms for restriction to Liouville subdomains.

Introduced new moduli spaces solving generalized continuation map equations.

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 Vitero. 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.