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Twisted generating functions and the nearby Lagrangian conjecture

Mohammed Abouzaid, Sylvain Courte, Stéphane Guillermou, Thomas Kragh

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 Bökstedt which computes the rational homotopy type of the tube space.

Twisted generating functions and the nearby Lagrangian conjecture

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 Bökstedt which computes the rational homotopy type of the tube space.

Paper Structure

This paper contains 29 sections, 51 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Gauss map for nearby Lagrangians
  3. Twisted generating functions
  4. Acknowledgements
  5. A twisted generalization of a theorem of Giroux and Latour
  6. Twisted generating functions
  7. Symplectic reduction and Bott periodicity
  8. Proof of Theorem \ref{['thmintro:twistedgirouxlatour']}
  9. "only if" part
  10. "if" part
  11. Step 1 : construction of an infinitesimal twisted generating function
  12. Step 2 : compatibility with a fixed embedding
  13. Step 3: from infinitesimal to actual functions
  14. The doubling trick and generating functions of tube type
  15. Twisted generating functions linear at infinity and the homotopy lifting property
  16. ...and 14 more sections

Key Result

Theorem A

Let $M$ be a closed manifold and $L$ a closed exact Lagrangian submanifold of $T^* M$, then the stable Gauss map $L\to \Lambda_0(\mathbf{C}^\infty)$ vanishes on all homotopy groups.

Figures (1)

  • Figure 1: The graph of the function $D$

Theorems & Definitions (130)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Remark 1.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Giroux, Latour
  • ...and 120 more