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 $\mathbb R^{2n}$ that equals $\mathbb R^n$ outside a compact set. Such a Lagrangian may be viewed as a Lagrangian filling of the standard Legendrian unknot $S^{n-1}$ in $D^{2n}$. 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 study of exact Lagrangians (and Legendrians) to algebraic K-theory of spaces. Using this and B\"okstedt's result that the Hatcher--Waldhausen map is a rational homotopy equivalence, we prove that the stable Lagrangian Gauss map (relative to the boundary) of the Lagrangian is null-homotopic.