A Beginner’s Introduction to Fukaya Categories

@article{Auroux2014ABI,
  title={A Beginner’s Introduction to Fukaya Categories},
  author={Denis Auroux},
  journal={arXiv: Symplectic Geometry},
  year={2014},
  pages={85-136}
}
  • D. Auroux
  • Published 29 January 2013
  • Mathematics
  • arXiv: Symplectic Geometry
The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce the Fukaya category (informally and without a lot of the necessary technical detail), and briefly discuss algebraic concepts such as exact triangles and generators. Finally, we mention wrapped Fukaya categories and outline a few applications to symplectic… 
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References

SHOWING 1-10 OF 52 REFERENCES
Fukaya categories and deformations
This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing
A geometric criterion for generating the Fukaya category
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
The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint
We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the
Fukaya categories of symmetric products and bordered Heegaard-Floer homology
The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and
Fukaya Categories as Categorical Morse Homology
The Fukaya category of a Weinstein manifold is an intricate symplectic inva- riant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in
A topological model for the Fukaya categories of plumbings
We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the
Fukaya categories and bordered Heegaard-Floer homology
We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent
On the wrapped Fukaya category and based loops
Given an exact relatively Pin Lagrangian embedding Q in a symplectic manifold M, we construct an A-infinity restriction functor from the wrapped Fukaya category of M to the category of modules on the
Fukaya A∞-structures Associated to Lefschetz Fibrations. II
Consider the Fukaya category associated to a Lefschetz fibration. It turns out that the Floer cohomology of the monodromy around ∞ gives rise to natural transformations from the Serre functor to the
An open string analogue of Viterbo functoriality
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
...
1
2
3
4
5
...