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Peer-reviewedCC BYInventiones mathematicae

Generating functions in ℝ2n and the Hatcher–Waldhausen map

  • Thomas Kragh1
  1. 1Department of Mathematics, Uppsala University, Uppsala, Sweden
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Abstract

In this paper, we construct a generating function quadratic at infinity for any exact Lagrangian in ℝ2n R 2 n that equals ℝn R n outside a compact set. Such a Lagrangian may be viewed as a Lagrangian filling of the standard Legendrian unknot Sn−1 S n − 1 in D2n 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.

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Cite

BibTeXRIS
BibTeX
@article{kragh2026generating,
  title = {{Generating functions in $\mathbbR^2n$ and the Hatcher–Waldhausen map}},
  author = {Thomas Kragh},
  journal = {Inventiones mathematicae},
  year = {2026},
  month = may,
  doi = {10.1007/s00222-026-01428-2},
  url = {https://doi.org/10.1007/s00222-026-01428-2},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Generating functions in ℝ^2n and the Hatcher–Waldhausen map
AU  - Thomas Kragh
PY  - 2026
DA  - 2026/05/18
JO  - Inventiones mathematicae
DO  - 10.1007/s00222-026-01428-2
UR  - https://doi.org/10.1007/s00222-026-01428-2
AB  - 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.
SN  - 0020-9910
SN  - 1432-1297
ER  -