Twisted generating functions and the nearby Lagrangian conjecture

TL;DR

This paper introduces twisted generating functions to analyze the homotopy properties of Lagrangian submanifolds in cotangent bundles, proving the null-homotopy of the stable Lagrangian Gauss map for spheres.

Contribution

It develops the concept of twisted generating functions and extends sheaf-theoretic methods to study the homotopy groups of Lagrangian submanifolds, linking to Waldhausen's tube space.

Findings

Homomorphism of homotopy groups vanishes for closed exact embedded Lagrangians.

The stable Lagrangian Gauss map is null-homotopic for all spheres.

The use of twisted generating functions constrains the topology of Lagrangian submanifolds.

Abstract

We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all spheres. The key tool that we introduce in order to prove this is the notion of twisted generating function and we show that every closed exact Lagrangian can be described using such an object, by extending a doubling argument developed in the setting of sheaf theory. Floer theory and sheaf theory constrain the type of twisted generating functions that can appear to a class which is closely related to Waldhausen's tube space, and our main result follows by a theorem of B\"okstedt which computes the rational homotopy type of the tube space.