Generating Functions in ℝ²ⁿ and the Hatcher-Waldhausen map

2018/04/07 by Kragh, Thomas
#19L99 #53D12 #57N70 #57R17 #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1804.02557

Abstract

In this paper we construct a generating function quadratic at infinity for any exact Lagrangian in \mathbb R²ⁿ equal to \mathbb Rⁿ outside a compact set. This type of Lagrangian is equivalent to a Lagrangian filling in D²ⁿ of the standard Legendrian unknot S^n-1. Generating functions of the type we construct are related to the space \mathcal M_∞ considered by Eliashberg and Gromov. We also show that \mathcal M_∞ 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.

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