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

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 ${\mathcal{M}}_{\mathrm{\infty }}$ considered by Eliashberg and Gromov. We also show that ${\mathcal{M}}_{\mathrm{\infty }}$ 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.