Table of contents

Table of contents for Polyfold Constructions for Fukaya Categories

Construction overviews:

• Moduli spaces of pseudoholomorphic polygons
• Regularized moduli spaces
• summary of resulting Floer chain complex (TODO)

Algebra details:

• Novikov ring (TODO)
• Brane structure (TODO)
  • Grading induced by brane structures (TODO)
• Orientations of Cauchy-Riemann sections induced by brane structures (TODO)

Geometry/Topology/Combinatorics details:

• Deligne-Mumford space (N-work-in-progress)
  • glued surface (N-work in progress -- and need to synchronize with reference to this in [expansion of expansion in point 7] and Gluing construction for Hamiltonians )
• Compactified Morse trajectory spaces (brief summary with references - could use extension)
• Coherent orientations on the regularized moduli spaces arising from polyfold theory (TODO)
• the polyfold ambient space ${\mathcal {X}}(\underline {x})$ as a topological space (K-TODO)
  • Gromov topology (K-TODO)
• the polyfold ambient bundle $\pi :{\mathcal {Y}}_{J}(\underline {x})$ as continuous surjection between topological spaces (K-TODO)
• the Cauchy-Riemann section as continuous map $\overline \partial _{{J,Y}}:{\mathcal {X}}(\underline {x})\to {\mathcal {Y}}_{J}(\underline {x})$ (K-TODO)
  • Gluing construction for Hamiltonians (TODO)

Analysis details:

• the polyfold smooth structure on the ambient space ${\mathcal {X}}(\underline {x})$ (J-TODO)
• the polyfold bundle structure of the ambient bundle $\pi :{\mathcal {Y}}_{J}(\underline {x})$ (J-TODO)
• the polyfold Fredholm property of the Cauchy-Riemann section $\overline \partial _{{J,Y}}:{\mathcal {X}}(\underline {x})\to {\mathcal {Y}}_{J}(\underline {x})$ (J-TODO)
• proof that Gromov compactness implies properness of the Cauchy-Riemann section (TODO)