[Paper Review] Symplectormophism groups of non-compact manifolds, orbifold balls, and a space of Lagrangians

Richard Hind, Martin Pinsonnault+1 more|arXiv (Cornell University)|May 31, 2013

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper establishes a homotopy equivalence between the space of symplectic embeddings of singular balls with conical singularities and the Kähler isometry group of Hirzebruch surfaces, proving that the space of Lagrangian real projective planes in the cotangent bundle of RP² is weakly contractible. Using symplectization and contact geometry, it connects symplectomorphism groups of non-compact manifolds to isometry groups of algebraic surfaces, extending known results on symplectic packing and Lagrangian spaces.

ABSTRACT

We establish connections between contact isometry groups of certain contact manifolds and compactly supported symplectomorphism groups of their symplectizations. We apply these results to investigate the space of symplectic embeddings of balls with a single conical singularity at the origin. Using similar ideas, we also prove the longstanding expected result that the space of Lagrangian $\RR P^2$ in $T^*\RR P^2$ is weakly contractible.

Motivation & Objective

• To understand the homotopy type of symplectomorphism groups of non-compact symplectic manifolds arising as symplectizations of contact manifolds.
• To analyze the space of symplectic embeddings of singular balls with conical singularities using geometric and topological techniques.
• To prove that the space of Lagrangian real projective planes in the cotangent bundle of RP² is weakly contractible, extending known results on Lagrangian spheres.
• To establish a correspondence between compactly supported symplectomorphisms of symplectized lens spaces and based loop spaces of isometry groups of algebraic surfaces.

Proposed method

• Reduces the symplectomorphism group of the symplectization of L(n,1) to the symplectomorphism group of the punctured total space of the line bundle O(n) minus its zero section.
• Uses the identification of sL(n,1) with the complement of the infinity section and zero section in the Hirzebruch surface F_n to relate symplectomorphism groups to Kähler isometry groups.
• Applies the theory of symplectic packing and isotopy to show that symplectic embeddings of singular balls are homotopy equivalent to the Kähler isometry group K_n of F_n.
• Employs a line field construction on a singular ball to relate compactly supported diffeomorphisms to sections of a trivialized projectivized tangent bundle.
• Uses the contractibility of spaces of diffeomorphisms on intervals and the contractibility of spaces of maps to R to show that the space of line fields homotopic to the trivial field is contractible.
• Establishes a homotopy equivalence between the space of compactly supported diffeomorphisms of a singular ball and the space of such line fields, proving the contractibility of the diffeomorphism group.

Experimental results

Research questions

• RQ1 What is the homotopy type of the group of compactly supported symplectomorphisms of the symplectization of a lens space L(n,1)?
• RQ2 How do symplectic embeddings of singular balls with conical singularities of order n relate to isometry groups of algebraic surfaces?
• RQ3 Is the space of Lagrangian RP² submanifolds in T*RP² weakly contractible, and how does this relate to known results on Lagrangian spheres?
• RQ4 Can the symplectomorphism group of a singular ball of size 1 be shown to be contractible via geometric and topological reduction?
• RQ5 What is the role of the Kähler isometry group of the Hirzebruch surface F_n in classifying symplectic embeddings of singular balls?

Key findings

• The group of compactly supported symplectomorphisms of the symplectization of L(n,1) is weakly homotopy equivalent to the based loop space of SU(2), with countably many components.
• The space of symplectic embeddings of a singular ball of size ε ∈ (0,1) into a singular ball of size 1 is homotopy equivalent to the Kähler isometry group K_n of the Hirzebruch surface F_n.
• The group of reduced, compactly supported symplectomorphisms of a singular ball of size 1 is contractible.
• The space of Lagrangian RP² submanifolds in T*RP², equipped with the C^∞-topology, is weakly contractible.
• For n=1, the result reduces to the known fact that the space of symplectic embeddings B(ε)→B(1) deformation retracts onto U(2).
• The space of compactly supported diffeomorphisms of a singular 4-ball with a conical singularity of order n is contractible, as shown via line field and isotopy techniques.