Introduction to sutured Floer homology

TL;DR

This paper provides a comprehensive, accessible introduction to sutured Floer homology, covering its construction, properties, and implications for graduate students in geometry and topology.

Contribution

It offers a clear, structured exposition connecting Lagrangian Floer homology, Heegaard Floer homology, and sutured Floer homology, including foundational background and key properties.

Findings

Sutured Floer homology detects the product structure.

It behaves well under surface decompositions.

Its Euler characteristic is computable via Fox calculus.

Abstract

This article is a standalone introduction to sutured Floer homology for graduate students in geometry and topology. It is divided into three parts. The first part is an introductory level exposition of Lagrangian Floer homology. The second part is a construction of Heegaard Floer homology as a special, and slightly modified, case of Lagrangian Floer homology. The third part covers the background on sutured manifolds, the definition of sutured Floer homology, as well as a discussion of its most basic properties and implications (it detects the product, behaves nicely under surface decompositions, defines an asymmetric polytope, its Euler characteristic is computable using Fox calculus).