A stable homotopy invariant for Legendrians with generating families

TL;DR

This paper introduces a stable homotopy type invariant for Legendrian submanifolds with generating families, providing new topological and algebraic constraints on Lagrangian fillings and generating family constructions.

Contribution

It constructs a spectrum-level invariant that lifts generating family homology and extends Seidel's isomorphism to the spectrum level, offering novel tools for Legendrian and Lagrangian topology.

Findings

Establishes topological constraints on Lagrangian fillings from generating families.

Provides algebraic constraints on the existence of generating family fillings.

Derives lower bounds on fiber dimensions needed for 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.