[Paper Review] Cluster Homology

Octav Cornea, François Lalonde|arXiv (Cornell University)|Aug 18, 2005

Geometric and Algebraic TopologyMathematics13 citations

TL;DR

This paper introduces a new homology invariant for Lagrangian submanifolds, called cluster homology, which controls disk bubbling through auxiliary Morse data. It provides a universal framework for Lagrangian intersection Floer theory and is invariant under Hamiltonian isotopy.

ABSTRACT

We assign, to a Langrangian submanifold $L$, a new homology which manages the bubbling of disks by means of auxiliary Morse data. This invariant of the Hamiltonian isotopy class of $L$ has many applications and naturally leads to a universal Floer theory for Lagrangian intersections.

Motivation & Objective

• To define a new homology theory for Lagrangian submanifolds that remains invariant under Hamiltonian isotopy.
• To address the issue of disk bubbling in Lagrangian Floer homology using auxiliary Morse data.
• To construct a universal framework for Floer theory of Lagrangian intersections.
• To provide a systematic tool for studying intersections of Lagrangians in symplectic manifolds.

Proposed method

• The construction uses Morse-Bott techniques to manage the compactness issues arising from holomorphic disk bubbling.
• Auxiliary Morse functions are introduced on the Lagrangian to stabilize the moduli space of holomorphic disks.
• The homology is defined as the Morse homology of a chain complex built from intersection points and holomorphic disks.
• The theory is shown to be invariant under Hamiltonian isotopy by controlling perturbations via Morse data.
• The construction naturally extends to a universal Floer theory for multiple Lagrangians.
• The framework allows for a consistent treatment of Lagrangian intersections even in the presence of bubbling.

Experimental results

Research questions

• RQ1 How can disk bubbling in Lagrangian Floer homology be systematically controlled to ensure compactness?
• RQ2 What structure can be assigned to a Lagrangian submanifold that remains invariant under Hamiltonian isotopy despite bubbling?
• RQ3 Can a universal Floer theory for Lagrangian intersections be constructed using auxiliary data?
• RQ4 How does the inclusion of Morse data refine the Floer complex in the presence of holomorphic disks?
• RQ5 What is the relationship between Morse-theoretic data and the invariance of the resulting homology?

Key findings

• The paper constructs a new homology invariant, cluster homology, which is preserved under Hamiltonian isotopy of the Lagrangian submanifold.
• Disk bubbling is managed through the use of auxiliary Morse functions, ensuring compactness of the moduli space.
• The resulting homology provides a universal framework for Lagrangian intersection Floer theory.
• The construction is robust under perturbations and yields a well-defined invariant in symplectic topology.
• The method establishes a bridge between Morse theory and Floer homology in the context of Lagrangian intersections.
• The theory offers a systematic approach to studying intersections even when bubbling obstructs standard Floer-theoretic constructions.