A stable homotopy invariant for Legendrians with generating families
Abstract
We construct a stable homotopy type invariant for any Legendrian submanifold in a jet bundle equipped with a linear-at-infinity generating family. We show that this spectrum lifts the generating family homology groups. When the generating family extends to a generating family for an embedded Lagrangian filling, we lift the Seidel isomorphism to the spectrum level. As applications, we establish topological constraints on Lagrangian fillings arising from generating families, algebraic constraints on whether generating families admit fillings, and lower bounds on how many fiber dimensions are needed to construct a generating family for a Legendrian.
Cite
@article{arxiv.2408.01587,
title = {A stable homotopy invariant for Legendrians with generating families},
author = {Hiro Lee Tanaka and Lisa Traynor},
journal= {arXiv preprint arXiv:2408.01587},
year = {2025}
}
Comments
Version to appear in Journal of Symplectic Geometry. Following referee suggestions: Typos corrected, Section 1.7 added for context and history, Example 1.11 generalized to a family of Legendrian knots containing m(5_2), a cofibration argument (Lemma 5.18) added, appendix on homotopy theory of spaces removed. 77 pages, 6 figures