# 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} }

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…

## 77 Citations

A Quick View of Lagrangian Floer Homology

- Mathematics
- 2018

In this note we present a brief introduction to Lagrangian Floer homology and its relation to the solution to the Arnol’d Conjecture, on the minimal number of non-degenerate fixed points of a…

Generalizations of Hodge-de-Rham degeneration for Fukaya categories

- Mathematics
- 2020

Above, δ denotes the Connes operator, reviewed in Section 2.2.1, and ch is the noncommutative Chern character, reviewed in Section 3.2.2. Symplectic geometry is a source of particularly interesting…

The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology

- Mathematics
- 2018

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We…

The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer theory in the pillowcase

- Mathematics
- 2014

We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram…

FLOER COHOMOLOGY AND FUKAYA CATEGORY

- Mathematics
- 2020

This paper aims to give an introduction to Floer Theory and relevant topics such as Arnold’s conjecture and Fukaya categories. We mostly follow the survey [1]. More details on the proofs and examples…

INTRODUCTION TO HOMOLOGICAL INVARIANTS IN LOW-DIMENSIONAL TOPOLOGY

- Mathematics
- 2017

There is a common main idea in various constructions of low-dimensional topological invariants. One takes a topological object, associates a geometric construction to it, which involves some choices,…

On the Fukaya category of a Fano hypersurface in projective space

- Mathematics
- 2016

This paper is about the Fukaya category of a Fano hypersurface X⊂CPn$X \subset \mathbf {CP}^{n}$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the…

Review of A ∞ Categories , Twisted Complexes , and Triangles

- Mathematics
- 2019

Let’s look at these composition maps (up to signs): If d = 1, then we have that μ(μ(a1)) = 0. In fact, we will form the cohomological category of A, where μ1 will be the differential. If d = 2, we…

for mini-course on the Fukaya category 1

- Mathematics
- 2016

We did not cover the analytical foundations (transversality, Gromov compactness, gluing) of Floer theory in the lecture. Here are some helpful references for learning this material: [MS04], [Sal99],…

Lefschetz fibrations on adjoint orbits

- Mathematics
- 2013

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular…

## References

SHOWING 1-10 OF 52 REFERENCES

Fukaya categories and deformations

- Mathematics
- 2002

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

- Mathematics
- 2010

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

- Mathematics
- 2008

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

- Mathematics
- 2010

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

- Mathematics
- 2014

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

- Mathematics
- 2009

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

- Mathematics
- 2010

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

- Mathematics
- 2009

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

- Mathematics
- 2014

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

- Mathematics
- 2007

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…