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An open string analogue of Viterbo functoriality

Mohammed Abouzaid and Paul Seidel

Geometry & Topology 14 (2010) 627–718

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.

Lagrangian Floer homology
Mathematical Subject Classification 2000
Primary: 53D40
Received: 8 January 2008
Revised: 15 August 2009
Accepted: 12 October 2009
Published: 16 February 2010
Proposed: Yasha Eliashberg
Seconded: Simon Donaldson, Leonid Polterovich
Mohammed Abouzaid
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Ave
Cambridge MA 02139
Paul Seidel
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Ave
Cambridge MA 02139