Abstract
We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler’s program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse–Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its symplectic neighborhood. We then study the singularities of the skeleta in this class and show that after a certain type of generic perturbation either (1) these singularities fall into the class of (signed Lagrangian versions of) Nadler’s arboreal singularities which are combinatorially classified into finitely many types in a given dimension or (2) there are singularities of tangency in associated front projections. We then turn to the singularities of tangency to try to reduce them also to collections of arboreal singularities. We give a general localization procedure to isolate the Liouville flow to a neighborhood of these non-arboreal singularities, and then show how to replace the simplest singularities of tangency (those of Thom-Boardman type \(\Sigma ^{1,0}\)) by arboreal singularities.
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References
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Acknowledgements
This work has greatly benefited from many discussions with David Nadler and Yasha Eliashberg. I am grateful for David’s invaluable intuition on arborealization which confirmed throughout when things were on track and corrected them when they were not. I have learned an enormous amount from Yasha and every discussion we have had has taught me a new way of thinking about Lagrangians, symplectic manifolds, and singularities. I have tried to incorporate some of these perspectives into my definitions and proofs, which I believe has significantly advanced the clarity and scope of these results. I am also grateful for advice, interest, shared knowledge, and suggestions from Daniel Álvarez-Gavela, Roger Casals, Kai Cieleibak, Josh Sabloff, Vivek Shende, and Alex Zorn. During the course of this work, I have been supported by an NSF Postdoctoral Fellowship Grant No. 1501728.
