[Paper Review] Reduction for quasi-morphisms on contactomorphism groups and contact rigidity

Matthew Strom Borman, Frol Zapolsky|arXiv (Cornell University)|Aug 14, 2013

Geometric and Algebraic Topology51 references3 citations

TL;DR

This paper constructs homogeneous quasi-morphisms on the universal cover of contactomorphism groups for prequantizations of symplectic toric manifolds using Givental’s nonlinear Maslov index and a novel contact reduction technique. The key contribution is a hierarchy of rigid subsets in contact manifolds, leading to symplectic rigidity for the real part of weighted projective spaces and new results on contact orderability and Sandon-type metrics.

ABSTRACT

We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. As a corollary we establish symplectic rigidity for the real part of weighted projective spaces. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.

Motivation & Objective

• To develop a method for constructing homogeneous quasi-morphisms on the universal cover of contactomorphism groups for specific prequantized symplectic toric manifolds.
• To establish a hierarchy of rigid subsets in contact manifolds using these quasi-morphisms.
• To prove symplectic rigidity for the real part of weighted projective spaces through the constructed invariants.
• To demonstrate a vanishing property of the nonlinear Maslov index, essential for the construction.
• To apply the results to contact orderability and the geometry of Sandon-type metrics on contactomorphism groups.

Proposed method

• Utilizes Givental’s nonlinear Maslov index as a foundational invariant for constructing quasi-morphisms on contactomorphism groups.
• Applies a contact reduction technique to quasi-morphisms, enabling the transfer of invariants from higher-dimensional contact manifolds to lower-dimensional ones.
• Employs the universal cover of the contactomorphism group to ensure the quasi-morphisms are homogeneous.
• Leverages the vanishing property of the nonlinear Maslov index to control behavior under reduction and ensure consistency.
• Combines symplectic and contact geometric techniques to analyze rigidity and metric structures.
• Uses the resulting quasi-morphisms to probe orderability and define metrics on contactomorphism groups.

Experimental results

Research questions

• RQ1 How can homogeneous quasi-morphisms be systematically constructed on the universal cover of contactomorphism groups for prequantized symplectic toric manifolds?
• RQ2 What role does the nonlinear Maslov index play in generating such quasi-morphisms, and does it satisfy a vanishing property under reduction?
• RQ3 Can these quasi-morphisms be used to define a hierarchy of rigid subsets in contact manifolds?
• RQ4 To what extent does this framework imply symplectic rigidity for the real part of weighted projective spaces?
• RQ5 How do these constructions influence the orderability of contact manifolds and the structure of Sandon-type metrics?

Key findings

• The construction of homogeneous quasi-morphisms on the universal cover of contactomorphism groups is achieved via contact reduction and Givental’s nonlinear Maslov index.
• A vanishing property of the nonlinear Maslov index is established, which is crucial for the consistency and behavior of the quasi-morphisms under reduction.
• The quasi-morphisms generate a hierarchy of rigid subsets in contact manifolds, reflecting deep geometric constraints.
• Symplectic rigidity is proven for the real part of weighted projective spaces using the constructed invariants.
• The results lead to new insights into the orderability of contact manifolds and the structure of Sandon-type metrics on contactomorphism groups.
• The framework provides a systematic method to relate geometric invariants in contact and symplectic topology through reduction techniques.