Generating functions in R^2n and the Hatcher–Waldhausen map

Abstract In this paper, we construct a generating function quadratic at infinity for any exact Lagrangian in R^2n R 2 n that equals R^n R n outside a compact set. Such a Lagrangian may be viewed as a Lagrangian filling of the standard Legendrian unknot S^n-1 S n − 1 in D^2n D 2 n. Generating functions of the type we construct are related to the space M M ∞ considered by Eliashberg and Gromov. We also show that M M ∞ 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 Bökstedt’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.