Holomorphic disks and three-manifold invariants: properties and applications

TL;DR

This paper studies Floer homology invariants for three-manifolds, exploring their properties, calculations, and applications, and suggests a connection with Seiberg-Witten theory.

Contribution

It provides detailed calculations and properties of Floer homology invariants, and proposes a conjectured link with Seiberg-Witten theory.

Findings

Relationship between Euler characteristics and Turaev's torsion

Connection with the Thurston norm and minimal genus problem

Surgery exact sequences for these invariants

Abstract

In an earlier paper (math.SG/0101206), we introduced Floer homology theories associated to closed, oriented three-manifolds Y and SpinC structures. In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with Seiberg-Witten theory. The properties include a relationship between the Euler characteristics of these theories and Turaev's torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology.