[Paper Review] Arborealization I: Stability of arboreal models

Daniel Álvarez‐Gavela, Yakov Eliashberg+1 more|arXiv (Cornell University)|Jan 12, 2021

Homotopy and Cohomology in Algebraic Topology9 references4 citations

TL;DR

This paper establishes a stability result for canonical models of arboreal singularities, showing they arise as closures of smooth Lagrangian germs under iterated transverse Liouville cones. It further proves that the space of symplectomorphisms preserving such models is weakly homotopy equivalent to the automorphism group of the associated signed rooted tree, reducing local symplectic topology to combinatorics.

ABSTRACT

We establish a stability result for canonical models of arboreal singularities. As a main application, we give a geometric characterization of the canonical models as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. The parametric version of the stability result implies that the space of germs of symplectomorphisms that preserve a canonical model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically.

Motivation & Objective

• To establish a stability result for canonical models of arboreal singularities.
• To provide a geometric characterization of canonical models as closures of smooth Lagrangian germs under iterated transverse Liouville cones.
• To show that the space of symplectomorphisms preserving a canonical model is weakly homotopy equivalent to the automorphism group of the corresponding signed rooted tree.
• To demonstrate that local symplectic topology around canonical models reduces to combinatorial data, even in the parametric setting.

Proposed method

• Use of iterated transverse Liouville cones to construct canonical models from smooth Lagrangian germs.
• Application of stability theorems in the context of arboreal singularities to ensure robustness under small perturbations.
• Identification of the symplectic automorphism space of a canonical model with the automorphism group of a signed rooted tree.
• Employment of parametric versions of stability results to extend homotopy equivalence to families of symplectomorphisms.
• Use of canonical models as limits of smooth Lagrangian germs to define geometric closure properties.
• Reduction of symplectic topology to combinatorics via the tree structure associated with the singularity.

Experimental results

Research questions

• RQ1 How do canonical models of arboreal singularities behave under small perturbations or deformations?
• RQ2 What is the geometric construction of canonical models in terms of Lagrangian germs and Liouville cones?
• RQ3 To what extent does the space of symplectomorphisms preserving a canonical model reflect the combinatorics of its associated signed rooted tree?
• RQ4 Can the local symplectic topology of a canonical model be fully captured by its combinatorial data?
• RQ5 How does the parametric version of the stability result affect the homotopy type of symplectomorphism spaces?

Key findings

• The canonical models of arboreal singularities are stable under small deformations, ensuring robustness of their geometric structure.
• Canonical models are geometrically characterized as the closure of smooth Lagrangian germs under iterated transverse Liouville cones.
• The space of symplectomorphisms preserving a canonical model is weakly homotopy equivalent to the automorphism group of the corresponding signed rooted tree.
• This homotopy equivalence holds even in the parametric setting, extending the result to families of symplectomorphisms.
• The local symplectic topology around a canonical model is fully determined by the combinatorics of its associated signed rooted tree.
• The reduction of symplectic topology to combinatorics provides a complete topological invariant for the local structure of arboreal singularities.