Fukaya Categories and Intersection Numbers

Abstract

We give an introductory survey of Floer homology and Fukaya categories assuming only basic symplectic geometry. The survey is meant to be targeted at a lower level than Auroux’s survey [3]. We say very little about the analytical details involved in Fukaya categories and move quickly to discuss the algebraic side. We show an application of these algebraic techniques by proving a theorem by Keating about symplectic Dehn twists [14]. This theorem is a generalization of the theorem that if two curves α, β have minimal geometric intersection number ≥ 2, then the Dehn twists τα, τβ generate a free subgroup of the mapping class group.