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Generating Functions in $\mathbb{R}^{2n}$ and the Hatcher-Waldhausen map

Thomas Kragh

Abstract

In this paper we construct a generating function quadratic at infinity for any exact Lagrangian in $\mathbb R^{2n}$ equal to $\mathbb R^n$ outside a compact set. This type of Lagrangian is equivalent to a Lagrangian filling in $D^{2n}$ of the standard Legendrian unknot $S^{n-1}$. Generating functions of the type we construct are related to the space $\mathcal M_\infty$ considered by Eliashberg and Gromov. We also show that $\mathcal M_\infty$ is the homotopy fiber of the so-called Hatcher-Waldhausen map. This further relates the understanding of exact Lagrangians (and Legendrians) to algebraic K-theory of spaces. As a result of this and the result by Bökstedt that the Hatcher-Waldhausen map is a rational homotopy equivalence we prove that the stable Lagrangian Gauss map (relative boundary) of the Lagrangian is homotopy trivial.

Generating Functions in $\mathbb{R}^{2n}$ and the Hatcher-Waldhausen map

Abstract

In this paper we construct a generating function quadratic at infinity for any exact Lagrangian in equal to outside a compact set. This type of Lagrangian is equivalent to a Lagrangian filling in of the standard Legendrian unknot . Generating functions of the type we construct are related to the space considered by Eliashberg and Gromov. We also show that is the homotopy fiber of the so-called Hatcher-Waldhausen map. This further relates the understanding of exact Lagrangians (and Legendrians) to algebraic K-theory of spaces. As a result of this and the result by Bökstedt that the Hatcher-Waldhausen map is a rational homotopy equivalence we prove that the stable Lagrangian Gauss map (relative boundary) of the Lagrangian is homotopy trivial.

Paper Structure

This paper contains 15 sections, 44 theorems, 165 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. A Hamiltonian moving part of the zero-section to $L$
  4. Cutting and pasting generating functions and proof of Theorem \ref{['thm:1']}.
  5. Extending to $n\leq 4$.
  6. The Space $\mathcal{M}_\infty$ and the Lagrangian Gauss-map
  7. The Fibration Sequence
  8. Parametrized handle attachments
  9. Identification of $\mathcal{U}_\infty \simeq F/O$.
  10. Identification of the Hatcher-Waldhausen map
  11. Level-wise construction
  12. Stabilizations
  13. Lifts
  14. Proof of Theorem \ref{['thm:3']}
  15. Homotopy groups of $\mathcal{M}_\infty$

Key Result

Theorem \oldthetheorem

Any exact Lagrangian $L\subset T^*D^n$ agreeing with the zero-section over a neighborhood of the boundary $\partial D^n=S^{n-1}$ has a generating function quadratic at infinity.

Figures (9)

  • Figure 1: The image of $i$ (red) and the smoothing defining $\mathop{\mathrm{hw}}\nolimits$ (blue).
  • Figure 2: Image of rotated zero-section for $n=1$
  • Figure 3: Image of rotated zero-section with compact support
  • Figure 4: Image of bumped off rotated zero-section for $n=1$
  • Figure 5: Rotation in $(q,p)$ coordinates.
  • ...and 4 more figures

Theorems & Definitions (92)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 82 more