A beginner's introduction to Fukaya categories

TL;DR

This paper provides a beginner-friendly overview of Fukaya categories, covering foundational concepts, algebraic structures, and applications in symplectic topology, mirror symmetry, and low-dimensional topology.

Contribution

It offers an accessible introduction to Fukaya categories and their applications, bridging foundational Floer theory with advanced topics in symplectic geometry.

Findings

Introduces Fukaya categories informally and conceptually.

Summarizes key algebraic structures like exact triangles.

Highlights applications in mirror symmetry and topology.

Abstract

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 topology, mirror symmetry and low-dimensional topology. This text is based on a series of lectures given at a Summer School on Contact and Symplectic Topology at Universit\'e de Nantes in June 2011.