[Paper Review] Holomorphic curves and continuation maps in Liouville bundles

Yong‐Geun Oh, Hiro TANAKA|arXiv (Cornell University)|Mar 10, 2020

Geometric and Algebraic Topology13 references4 citations

TL;DR

This paper constructs an unwrapped Floer theory for bundles of Liouville manifolds and sectors, establishing a compatible system of unwrapped Fukaya categories over simplices and proving that two constructions of continuation maps—via once-punctured disks and strips—are chain-homotopic. This framework enables the construction of homotopically coherent actions of Lie groups on wrapped Fukaya categories, proving a conjecture of Teleman (2014) in the Liouville and monotone settings.

ABSTRACT

We construct an unwrapped Floer theory for bundles of Liouville sectors. In particular, we construct a compatible collection of unwrapped Fukaya categories of fibers of a Liouville bundle, and prove that the two natural constructions of continuation maps in this setting behave compatibly. These constructions are exploited in [OT19] to construct homotopically coherent actions of Lie groups on wrapped Fukaya categories, thereby proving a conjecture from Teleman's 2014 ICM address.

Motivation & Objective

• To develop a consistent unwrapped Floer theory for fiber bundles of Liouville manifolds and sectors.
• To define and study continuation maps in this geometric setting using holomorphic curves.
• To prove that two distinct constructions of continuation maps (via once-punctured disks and strips) are chain-homotopic.
• To set the foundation for constructing homotopically coherent actions of Lie groups on wrapped Fukaya categories.
• To provide a geometric framework for studying symmetries in symplectic topology via holomorphic curves in Liouville bundles.

Proposed method

• Constructs a family of unwrapped Fukaya categories over simplices mapping smoothly to the classifying space BG of a Lie group G.
• Uses holomorphic curves with boundary conditions in fibers and punctures to define continuation maps.
• Applies compactness and transversality techniques, including C⁰ and energy estimates, to control moduli spaces of pseudoholomorphic curves.
• Employs operadic and gluing techniques for families of disks with strip-like ends and marked points.
• Utilizes one-jet transversality to rule out bubbling in interior parameters and ensures well-defined moduli spaces.
• Applies the stable map topology scheme with minimal marked points to prove homotopy equivalence of continuation maps.

Experimental results

Research questions

• RQ1 How can unwrapped Floer theory be consistently defined for fiber bundles of Liouville manifolds and sectors?
• RQ2 What is the relationship between two natural constructions of continuation maps in this setting—via once-punctured disks and via strips?
• RQ3 Can the compatibility of these continuation maps be established via a chain-homotopy identity?
• RQ4 How can group actions on Liouville manifolds be encoded in Floer-theoretic invariants using holomorphic curves in bundles?
• RQ5 Can this framework realize homotopically coherent actions of Lie groups on wrapped Fukaya categories, as conjectured by Teleman?

Key findings

• The two constructions of continuation maps—via once-punctured disks and via strips—are proven to be chain-homotopic, establishing their compatibility.
• The moduli space of holomorphic curves with boundary conditions in fibers and punctures admits a compactification with controlled boundary strata, including contributions from lower-dimensional moduli spaces.
• The map $\mathcal{H}$ defined by counting 0-dimensional moduli spaces of holomorphic curves satisfies the identity $ h_{\mathcal{L}}^{\rho} - \mu^2(c^{\chi}_{\mathcal{L}}, \ast) = \mu^1 \mathcal{H} + \mathcal{H} \mu^1 $, proving the homotopy equivalence.
• The unwrapped Fukaya category over a simplex is well-defined and compatible with face inclusions, forming a coherent system over BG.
• The construction provides a geometric realization of homotopically coherent actions of Lie groups on wrapped Fukaya categories, confirming a conjecture of Teleman (2014).
• The framework avoids the need for strictly G-equivariant data by encoding symmetries via holomorphic curves in the Liouville bundle over BG.