Sectorial descent for wrapped Fukaya categories

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Abstract
We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful.
Funding Information
  • National Science Foundation (1907635)
  • National Science Foundation (1440140)
  • National Science Foundation (1747553)
  • National Science Foundation (NSF CAREER DMS-1654545)
  • National Science Foundation (1907635)
  • National Science Foundation (1440140)
  • National Science Foundation (1747553)
  • National Science Foundation (NSF CAREER DMS-1654545)
  • Clay Mathematics Institute (1907635)
  • Clay Mathematics Institute (1440140)
  • Clay Mathematics Institute (1747553)
  • Clay Mathematics Institute (NSF CAREER DMS-1654545)
  • David and Lucile Packard Foundation (1907635)
  • David and Lucile Packard Foundation (1440140)
  • David and Lucile Packard Foundation (1747553)
  • David and Lucile Packard Foundation (NSF CAREER DMS-1654545)
  • National Science Foundation (1907635)
  • National Science Foundation (1440140)
  • National Science Foundation (1747553)
  • National Science Foundation (NSF CAREER DMS-1654545)

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