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A geometric criterion for generating the Fukaya category

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Abstract

Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the identity in symplectic cohomology lies in the image of this map, we conclude that every Lagrangian lies in the idempotent closure of the chosen collection. The main new ingredients are (1) the construction of operations on the Fukaya category controlled by discs with two outputs, and (2) the Cardy relation.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA

    Mohammed Abouzaid

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  1. Mohammed Abouzaid
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Correspondence to Mohammed Abouzaid.

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This research was conducted during the period the author served as a Clay Research Fellow.

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Abouzaid, M. A geometric criterion for generating the Fukaya category. Publ.math.IHES 112, 191–240 (2010). https://doi.org/10.1007/s10240-010-0028-5

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  • DOI: https://doi.org/10.1007/s10240-010-0028-5

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