Lagrangian Cobordisms

Definition

Lagrangian cobordisms refer to a relationship between two Lagrangian submanifolds in a symplectic manifold, where these submanifolds can be connected by a compact Hamiltonian isotopy within a higher-dimensional symplectic manifold. This concept is crucial as it provides insights into the interactions and transformations between Lagrangian submanifolds, particularly when considering their intersections and how they can change under continuous deformations. It is tied closely to the study of topology and geometry, particularly in the context of Darboux's theorem, which ensures local normal forms for symplectic manifolds.

5 Must Know Facts For Your Next Test

1. Lagrangian cobordisms can be visualized as 'bridges' connecting two Lagrangian submanifolds within a higher-dimensional symplectic space.
2. In many cases, Lagrangian cobordisms help in studying the topology of Lagrangian submanifolds by analyzing their behaviors under various isotopies.
3. The existence of Lagrangian cobordisms has implications for Floer homology, which is used to define invariants that can differentiate between Lagrangian submanifolds.
4. Lagrangian cobordisms arise naturally when considering moduli spaces of pseudo-holomorphic curves in symplectic geometry.
5. Understanding Lagrangian cobordisms enhances the applications of Darboux's theorem by illustrating how local properties can lead to global topological effects.

Review Questions

• How do Lagrangian cobordisms facilitate connections between different Lagrangian submanifolds in symplectic geometry?
  • Lagrangian cobordisms act as continuous bridges that link two Lagrangian submanifolds within a larger symplectic manifold. They allow for the study of how these submanifolds can be transformed into one another through Hamiltonian isotopies. By examining these connections, mathematicians can gain insights into the geometric and topological properties that govern the behavior of these submanifolds over time.
• Discuss the role of Lagrangian cobordisms in relation to Floer homology and how it helps differentiate between various Lagrangian submanifolds.
  • Lagrangian cobordisms are essential for developing Floer homology, which provides invariants that can distinguish between different Lagrangian submanifolds. By connecting two Lagrangians through a cobordism, one can analyze the moduli space of pseudo-holomorphic curves that arise from this connection. The resulting invariants from Floer homology can help identify when two seemingly similar Lagrangians are actually distinct due to their topological characteristics.
• Evaluate the significance of Lagrangian cobordisms in enhancing our understanding of Darboux's theorem and its implications for symplectic geometry.
  • Lagrangian cobordisms deepen our comprehension of Darboux's theorem by providing a framework for relating local geometric properties to broader topological structures in symplectic geometry. As Darboux's theorem guarantees local normal forms for symplectic manifolds, Lagrangian cobordisms illustrate how these local forms can be connected in more complex ways across dimensions. This connection sheds light on how local transformations lead to significant global changes in the behavior and classification of Lagrangian submanifolds.

Related terms

Lagrangian Submanifold: A Lagrangian submanifold is a special type of submanifold in a symplectic manifold where the symplectic form restricts to zero, meaning that it has half the dimension of the ambient symplectic manifold.

Symplectic Manifold: A symplectic manifold is a smooth, even-dimensional differentiable manifold equipped with a closed non-degenerate 2-form called the symplectic form, which encodes geometric properties of the manifold.

Hamiltonian Isotopy: Hamiltonian isotopy is a continuous deformation of a Hamiltonian system that preserves the symplectic structure, allowing one to relate different states or configurations of a system through smooth transformations.