[Paper Review] Lagrangian Cobordism I

Paul Biran, Octav Cornea|arXiv (Cornell University)|Sep 23, 2011

Geometric and Algebraic Topology19 references4 citations

TL;DR

This paper introduces a categorical framework for monotone Lagrangian cobordism in symplectic topology, showing that Floer homology is preserved under cobordism and constructing a functor from a Lagrangian cobordism category to a triangulated Fukaya category. The key contribution is a functorial relationship between cobordism morphisms and Floer-theoretic invariants, establishing a bridge between geometric cobordisms and algebraic structures in symplectic topology.

ABSTRACT

This paper discusses the cobordism of Lagrangian submanifolds (in the monotone setting) and structures it as a category that is related in a functorial way to an appropriate (derived) Fukaya category. Are also discussed obstructions to cobordism based on properties of Lagrangian quantum homology, relations to Lagrangian surgery, as well as examples of non-isotopic but cobordant Lagrangians. This is a revision of our earlier preprint from September 2011.

Motivation & Objective

• To develop a categorical structure for monotone Lagrangian cobordisms in symplectic manifolds.
• To show that Floer homology and related invariants are preserved under monotone Lagrangian cobordisms.
• To construct a functor from the Lagrangian cobordism category to a triangulated Fukaya category.
• To establish algebraic obstructions to cobordism existence via quantum and Floer homology invariants.
• To provide a framework for understanding cobordism as a functorial extension of Fukaya category structures.

Proposed method

• Define monotone Lagrangian submanifolds via uniform monotonicity and Maslov index constraints with $N_L \geq 2$.
• Introduce Lagrangian cobordism as a smooth compact cobordism embedded in $\mathbb{R}^2 \times M$ with cylindrical ends matching Lagrangian families.
• Construct a triangulated category $T^S\mathcal{C}$ from a full subcategory of the derived Fukaya category $D\mathcal{F}uk^d(M)$ using cone decompositions.
• Define morphisms in the cobordism category $\mathcal{Cob}^d_0(M)$ via sequences of chain maps and cone constructions in Floer complexes.
• Use the Yoneda embedding and cone construction in chain complexes to build a triangulated completion of the Fukaya category.
• Define a monoidal functor $\widetilde{\mathcal{F}}$ from $\mathcal{Cob}^d_0(M)$ to $T^S\mathcal{F}uk^d(M)$, with morphisms realized as chain homotopy equivalences between Floer complexes.

Experimental results

Research questions

• RQ1 How can Lagrangian cobordism be structured as a category with functorial properties?
• RQ2 What invariants are preserved under monotone Lagrangian cobordisms?
• RQ3 Can a functorial relationship be established between the Lagrangian cobordism category and the triangulated Fukaya category?
• RQ4 What algebraic obstructions arise from quantum and Floer homology in the context of cobordism?
• RQ5 How does the choice of ground ring $K = \mathbb{Z}_2$ affect the functoriality and independence of the constructions?

Key findings

• Monotone Lagrangian cobordisms preserve Floer homology and related invariants, as shown by Theorem 2.2.1.
• The existence of a functor $\widetilde{\mathcal{F}}$ from the cobordism category to the triangulated Fukaya category is established via cone decompositions of Floer complexes.
• The construction relies on a triangulated completion of the Fukaya category using Yoneda embedding and cone operations in chain complexes.
• The key result is that morphisms in the cobordism category induce exact sequences of chain complexes $Z_i^N$ with $Z_{i+1}^N = \textnormal{cone}(CF(N,L_i) \xrightarrow{u_i} Z_i^N)$, leading to a chain homotopy equivalence $\phi_V^N: CF(N,L) \to Z_{k+1}^N$.
• The use of $\mathbb{Z}_2$ as the ground ring ensures that the maps $u_i$ and $\phi_V^N$ are canonical and independent of additional choices, unlike over $\mathcal{A}$.
• Theorems 2.2.2 and 2.2.3 reveal algebraic obstructions to the existence of cobordism morphisms via quantum and Floer homology invariants.