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Orientations of Morse flow trees in Legendrian contact homology

Cecilia Karlsson

Abstract

Let $Λ$ be a closed, connected, spin Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. We give a coherent orientation scheme for the moduli space of rigid Morse flow trees of $Λ$, implying that the Legendrian contact homology of $Λ$ with integer coefficients can be computed using Morse flow trees. If $n>1$ then this orientation scheme can be computed with an algorithm which uses intersections of oriented flow manifolds in $M$ together with combinatorial data coming from the trees.

Orientations of Morse flow trees in Legendrian contact homology

Abstract

Let be a closed, connected, spin Legendrian submanifold of the 1-jet space of a smooth -dimensional manifold. We give a coherent orientation scheme for the moduli space of rigid Morse flow trees of , implying that the Legendrian contact homology of with integer coefficients can be computed using Morse flow trees. If then this orientation scheme can be computed with an algorithm which uses intersections of oriented flow manifolds in together with combinatorial data coming from the trees.

Paper Structure

This paper contains 85 sections, 79 theorems, 610 equations, 28 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Organization of the paper
  4. Morse flow trees
  5. Flow lines and vertices
  6. Partial flow trees
  7. Description of a neighborhood of a sub flow tree
  8. Flow outs of some particular sub flow trees
  9. Standard domains
  10. Orientation of pseudo-holomorphic disks
  11. J-holomorphic disks
  12. Complex structure
  13. Punctured J-holomorphic disks and Lagrangian boundary conditions
  14. Weighted Sobolev spaces and Fredholm operators
  15. Boundary conditions close to punctures
  16. ...and 70 more sections

Key Result

Theorem 1.1

Let $\Lambda \subset J^1(M)$ be a closed, connected, spin Legendrian submanifold, and assume that we have fixed all initial orientation choices. Then there is a coherent orientation scheme for the moduli space of rigid pseudo-holomorphic disks of $\Lambda$ so that the following holds. Assume that $\

Figures (28)

  • 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: The front projection of 1-jet lifts of a gradient flow line.
  • Figure 3: 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 4: The flow-out along $\Gamma$.
  • Figure 5: A standard domain $\Delta_{m+1}$, $m = 5$.
  • ...and 23 more figures

Theorems & Definitions (218)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 208 more