[Paper Review] Hamiltonian loops on the symplectic blow up

Andrés Pedroza|arXiv (Cornell University)|Oct 6, 2015

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper demonstrates that a Hamiltonian loop on a symplectic manifold can be lifted to a Hamiltonian loop on its one-point symplectic blow-up. Using Weinstein's morphism, it proves the lifted loop has infinite order in the fundamental group of the Hamiltonian diffeomorphism group of the blown-up manifold, establishing a non-trivial topological invariant in symplectic topology.

ABSTRACT

We lift a Hamiltonian loop on a symplectic manifold to a Hamiltonian loop on the symplectic one-point blow up of a symplectic manifold. Then we use Weinstein's morphism to show that the lifted Hamiltonian loop has infinite order on the fundamental group of the group of Hamiltonian diffeomorphisms of the blown up manifold.

Motivation & Objective

• To investigate the behavior of Hamiltonian loops under symplectic blow-up operations.
• To determine whether the fundamental group of the Hamiltonian diffeomorphism group changes under one-point symplectic blow-up.
• To apply Weinstein's morphism to detect non-trivial topology in the lifted loop.
• To establish that the lifted loop has infinite order in the fundamental group of the Hamiltonian diffeomorphism group of the blown-up manifold.

Proposed method

• Lift a given Hamiltonian loop from the original symplectic manifold to its one-point symplectic blow-up using symplectic geometry techniques.
• Construct a lift that preserves the Hamiltonian nature of the loop on the blown-up manifold.
• Apply Weinstein's morphism, a homomorphism from the fundamental group of the Hamiltonian diffeomorphism group to the first homology group of the manifold.
• Use the morphism to analyze the image of the lifted loop and deduce its non-triviality.
• Show that the image under Weinstein's morphism is non-zero, implying the loop has infinite order.
• Conclude that the fundamental group of the Hamiltonian diffeomorphism group of the blown-up manifold contains a non-torsion element.

Experimental results

Research questions

• RQ1 Does the symplectic blow-up operation preserve or alter the fundamental group structure of the Hamiltonian diffeomorphism group?
• RQ2 Can a Hamiltonian loop on the original manifold be lifted to a non-contractible loop on the blown-up manifold?
• RQ3 What is the order of the lifted Hamiltonian loop in the fundamental group of the Hamiltonian diffeomorphism group of the blow-up?
• RQ4 How does Weinstein's morphism detect non-trivial topology in the lifted loop?
• RQ5 Is the lifted loop of infinite order in the fundamental group of the Hamiltonian diffeomorphism group of the symplectic blow-up?

Key findings

• The Hamiltonian loop on the original manifold lifts to a Hamiltonian loop on the one-point symplectic blow-up.
• The lifted loop is non-contractible in the fundamental group of the Hamiltonian diffeomorphism group of the blown-up manifold.
• Weinstein's morphism maps the lifted loop to a non-zero element in the first homology group.
• The non-vanishing image under Weinstein's morphism implies the lifted loop has infinite order.
• The fundamental group of the Hamiltonian diffeomorphism group of the blown-up manifold contains a non-torsion element.
• The result establishes a non-trivial topological invariant in the symplectic topology of blow-ups.