Generating Functions in $\mathbb{R}^{2n}$ and the Hatcher-Waldhausen map
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2018
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911628292784128 |
|---|---|
| author | Kragh, Thomas |
| author_facet | Kragh, Thomas |
| contents | 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 $\mathcal M_\infty$ considered by Eliashberg and Gromov. We also show that $\mathcal M_\infty$ 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_02557 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Generating Functions in $\mathbb{R}^{2n}$ and the Hatcher-Waldhausen map Kragh, Thomas Symplectic Geometry Algebraic Topology K-Theory and Homology 53D12, 57R17, 19L99, 57N70 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 $\mathcal M_\infty$ considered by Eliashberg and Gromov. We also show that $\mathcal M_\infty$ 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 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. |
| title | Generating Functions in $\mathbb{R}^{2n}$ and the Hatcher-Waldhausen map |
| topic | Symplectic Geometry Algebraic Topology K-Theory and Homology 53D12, 57R17, 19L99, 57N70 |
| url | https://arxiv.org/abs/1804.02557 |