Generating Families for Lagrangians in $mathbb{R}^{2n}$ and the Hatcher-Waldhausen map

In this paper we construct a generating family quadratic at infinity for any exact Lagrangian in $mathbb{R}^{2n}$ equal to $mathbb{R}^n$ outside a compact set. Such generating families are related to the space $mathcal{M}_infty$ considered by Eliashberg and Gromov. We show that this space is the homotopy fiber of the Hatcher-Waldhausen map, and thus serves as a geometric model for this space. This relates the understanding of exact Lagrangians (and Legendrians) to algebraic K-theory of spaces. We then use this fibration sequence to produce new results (restrictions) on this type of Lagrangian. In particular we show how Bokstedts result that the Hatcher-Waldhausen map is a rational homotopy equivalence proves the new result that the stable Lagrangian Gauss map for our Lagrangian relative infinity is homotopy trivial.