[Paper Review] Holomorphic disks and topological invariants for rational homology three-spheres

Peter Ozsváth, Zoltán Szabó|arXiv (Cornell University)|Jan 25, 2001

Geometric and Algebraic Topology31 references10 citations

TL;DR

This paper introduces new topological invariants for rational homology three-spheres using holomorphic disks in symmetric products of Heegaard surfaces. By adapting Lagrangian Floer homology to Spin C structures, it constructs Z-graded Abelian groups that provide invariants under the action of the mapping class group, offering a new framework for studying 3-manifold topology via symplectic geometry.

ABSTRACT

The aim of this article is to introduce and study certain topological invariants for oriented, rational homology three-spheres Y. These groups are relatively Z-graded Abelian groups associated to Spin C structures over Y. Given a Heegaard splitting of Y = U0 ∪Σ U1, these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U0 and U1.

Motivation & Objective

• To define new topological invariants for oriented rational homology three-spheres using symplectic topology.
• To associate Z-graded Abelian groups to Spin C structures on such 3-manifolds.
• To generalize Lagrangian Floer homology to the setting of Heegaard splittings of rational homology spheres.
• To establish invariants that are functorial under cobordisms and compatible with the action of the mapping class group.
• To provide a symplectic-geometric framework for studying 3-manifold invariants via holomorphic disks in symmetric products.

Proposed method

• Utilizes a Heegaard splitting Y = U₀ ∪Σ U₁ to decompose the 3-manifold into handlebodies.
• Constructs two totally real submanifolds in the g-fold symmetric product of the Heegaard surface Σ, corresponding to U₀ and U₁.
• Applies Lagrangian Floer homology to these submanifolds to define a homology theory.
• Endows the resulting homology groups with a relative Z-grading via the first Chern class of the Spin C structure.
• Uses holomorphic disks in the symmetric product as the primary objects to compute the differential in the Floer complex.
• Establishes invariance under isotopy of the Lagrangian submanifolds and under Heegaard moves.

Experimental results

Research questions

• RQ1 How can Lagrangian Floer homology be adapted to define invariants for rational homology three-spheres?
• RQ2 What is the role of Spin C structures in grading the resulting homology groups?
• RQ3 How do the invariants transform under topological operations such as Heegaard moves or cobordisms?
• RQ4 Can holomorphic disks in symmetric products of Heegaard surfaces yield computable invariants for 3-manifolds?
• RQ5 What is the relationship between the new invariants and classical invariants like Reidemeister torsion or Seiberg-Witten invariants?

Key findings

• The paper constructs a Z-graded Abelian group invariant for each Spin C structure on a rational homology three-sphere.
• The invariants are defined via holomorphic disks in the symmetric product of a Heegaard surface, using Lagrangian Floer homology.
• The resulting homology groups are independent of the choice of Heegaard splitting, up to canonical isomorphism.
• The construction is functorial under cobordisms and respects the action of the mapping class group.
• The invariants are non-trivial for non-trivial 3-manifolds and detect topological distinctions among rational homology spheres.
• The grading on the homology groups is determined by the first Chern class of the Spin C structure, providing a refined invariant.