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 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.

71 pages, 9 figures, minor revision correcting typographical errors and clarifying the exposition following a referee's comments

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• Gromov-Witten classes, quantum cohomology, and enumerative geometry
• Homogeneous coordinate rings and mirror symmetry for toric varieties
• Exact Lagrangian submanifolds in simply-connected cotangent bundles

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