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To compute orientations of Morse flow trees in Legendrian contact homology

Cecilia Karlsson

Abstract

Let $Λ$ be a closed, connected Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. Associated to $Λ$ there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of $Λ$. Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of $Λ$, which allows us to compute the Legendrian contact homology of $Λ$ via Morse theory. If $Λ$ is spin, then the moduli space of the rigid disks can be given a coherent orientation, so that the Legendrian contact homology of $Λ$ can be defined with coefficients in $\mathbb Z$. In this paper we give an algorithm for computing the corresponding orientation of the moduli space of rigid Morse flow trees if the dimension of $Λ$ is greater than 1, and up to 4 signs that depend on data that can be extracted from the vertices of the trees, but which are not given explicitly, in the case $n=1$.

To compute orientations of Morse flow trees in Legendrian contact homology

Abstract

Let be a closed, connected Legendrian submanifold of the 1-jet space of a smooth -dimensional manifold. Associated to there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of . Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of , which allows us to compute the Legendrian contact homology of via Morse theory. If is spin, then the moduli space of the rigid disks can be given a coherent orientation, so that the Legendrian contact homology of can be defined with coefficients in . In this paper we give an algorithm for computing the corresponding orientation of the moduli space of rigid Morse flow trees if the dimension of is greater than 1, and up to 4 signs that depend on data that can be extracted from the vertices of the trees, but which are not given explicitly, in the case .

Paper Structure

This paper contains 25 sections, 2 theorems, 68 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Morse flow trees
  3. Flow lines and vertices
  4. Correspondence between disks and trees
  5. Partial flow trees
  6. Flow-outs and intersection manifolds
  7. Flow outs of some particular sub flow trees
  8. The sign v
  9. Initial choices
  10. Trivializations
  11. Definition of v
  12. The sign v
  13. Intersection orientation
  14. Orientation of flow-outs and intersection manifolds
  15. Definition of v
  16. ...and 10 more sections

Key Result

Theorem \oldthetheorem

Let $\Lambda$ be a closed, connected, spin Legendrian submanifold of $J^1(M)$, and assume that we have fixed all initial orientation choices. Let $\mathcal{M}$ be the moduli space of rigid Morse flow trees of $\Lambda$. Then there is a coherent orientation scheme of $\mathcal{M}$ so that the sign $\ where $\nu_{\operatorname{triv}}(\Gamma)$ , $\nu_{\operatorname{int}}(\Gamma)$ and $\nu_{\operatorn

Figures (16)

  • Figure 1: The local picture of a Morse flow tree $\Gamma$ in a neighborhood of a $Y_0$-vertex. The graphs of the defining functions $f_1$, $f_2$ and $f_3$ are sketched, together with the lift of the tree to the sheets of $\Lambda$ determined by these functions. The shaded area indicates the corresponding pseudo-holomorphic disk.
  • Figure 2: A standard domain $\Delta_{m+1}$, $m = 5$.
  • Figure 3: A tree $\Gamma$ with one positive 1-valent puncture, 2 $Y_0$- / $Y_1$-vertices, one negative 2-valent puncture (or switch), and 3 negative 1-valent punctures /ends. To the right the corresponding subdivision into vertex regions (filled) and edge regions (dotted).
  • Figure 4: Local picture in the Lagrangian projection close to an end- switch- or $Y_1$-vertex, consisting of a bended one-dimensional sheet $W$ times $\mathbb{R}^{n-1}$.
  • Figure 5: Local coordinates at a cusp. Here $v$ represents a switch, end or $Y_1$-vertex, and $\Pi(\Lambda_v)$ is given the orientation from $\partial_{x_1}$.
  • ...and 11 more figures

Theorems & Definitions (28)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 18 more