[Paper Review] Floer Cohomology and Geometric Composition of Lagrangian Correspondences

Katrin Wehrheim, Chris Woodward|arXiv (Cornell University)|May 9, 2009

Geometric and Algebraic Topology15 references4 citations

TL;DR

This paper establishes an isomorphism between Floer cohomologies under geometric composition of Lagrangian correspondences in exact and monotone symplectic settings. Using quilted Floer cohomology and analysis of pseudoholomorphic strips, it proves that composing Lagrangian correspondences preserves Floer cohomology when the composition is embedded, generalizing the behavior of symplectomorphisms in Floer theory.

ABSTRACT

We prove an isomorphism of Floer cohomologies under geometric composition of Lagrangian correspondences in exact and monotone settings.

Motivation & Objective

• To establish a canonical isomorphism between Floer cohomology groups under geometric composition of Lagrangian correspondences.
• To extend the functoriality of Floer cohomology from symplectomorphisms to more general Lagrangian correspondences.
• To provide a foundational tool for symplectic topological invariants via quilted Floer homology in higher-dimensional cobordisms.
• To verify that the composition of Lagrangian correspondences preserves the algebraic structure of Floer cohomology under monotonicity or exactness conditions.
• To resolve the question of how Floer cohomology transforms under geometric composition, particularly in the embedded case.

Proposed method

• Uses quilted Floer cohomology to define Floer cohomology for cyclic sequences of Lagrangian correspondences between symplectic manifolds.
• Applies pseudoholomorphic strip counts with boundary conditions matching via Lagrangian correspondences in the product space.
• Imposes monotonicity or exactness conditions on Lagrangian pairs to ensure well-defined Floer cohomology groups.
• Employs a canonical bijection between intersection points of Lagrangians in the product space and those in the composed correspondence.
• Analyzes convergence of pseudoholomorphic strips using weighted Sobolev norms and cutoff functions to rule out energy concentration.
• Applies a generalized Hofer-type lemma to control pointwise convergence and derive contradiction from non-vanishing gradient norms.

Experimental results

Research questions

• RQ1 Does geometric composition of Lagrangian correspondences preserve Floer cohomology in the embedded case?
• RQ2 Under what conditions does the isomorphism between Floer cohomology groups hold after composition of Lagrangian correspondences?
• RQ3 How does the quilted Floer cohomology behave when two Lagrangian correspondences are composed into a single one?
• RQ4 Can the canonical bijection between intersection points induce a cohomological isomorphism under monotonicity or exactness?
• RQ5 What analytical conditions ensure that pseudoholomorphic strips do not develop energy concentration or bubble formation during composition?

Key findings

• An isomorphism is established between the Floer cohomology of a pair of Lagrangians in the product space and the cohomology of the composed Lagrangian correspondence.
• The isomorphism holds under monotonicity or exactness assumptions on the Lagrangian pair in the ambient product manifold.
• The minimal Maslov index is required to be 2 to exclude nontrivial holomorphic disks that could obstruct the isomorphism.
• The proof relies on uniform bounds in weighted Sobolev norms and compactness arguments to rule out bubble formation in the moduli space.
• Pointwise convergence of gradients of pseudoholomorphic strips is shown to vanish in the limit, contradicting the assumption of non-zero energy concentration.
• The canonical bijection between intersection points induces a chain-level isomorphism, which lifts to an isomorphism on cohomology.