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A formula for the Euler class of foliations

Alessandro V. Cigna

Abstract

Given a cooriented branched surface $\mathcal B$ fully carrying a foliation $\mathcal F$, we use the dual graph of $\mathcal B$ to define a simplicial 1-cycle $Γ(\mathcal B)$ representing the Poincaré dual of the Euler class of $\mathcal F$. This construction provides an effective way to track how the Euler class of a foliation carried by a hierarchy changes when the orientation of a decomposing disk is reversed. We also show how our formula generalises previous results of Lackenby and Dunfield.

A formula for the Euler class of foliations

Abstract

Given a cooriented branched surface fully carrying a foliation , we use the dual graph of to define a simplicial 1-cycle representing the Poincaré dual of the Euler class of . This construction provides an effective way to track how the Euler class of a foliation carried by a hierarchy changes when the orientation of a decomposing disk is reversed. We also show how our formula generalises previous results of Lackenby and Dunfield.
Paper Structure (9 sections, 16 theorems, 27 equations, 11 figures)

This paper contains 9 sections, 16 theorems, 27 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Sutured manifolds and branched surfaces
  3. Sutured manifolds and their hierarchies
  4. Branched surfaces.
  5. From sutured manifold hierarchies to branched surfaces.
  6. Euler classes of plane bundles
  7. The maw dual graph
  8. Producing nonisotopic foliations
  9. Previous work on graphs and taut foliations

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a taut foliation of a compact oriented $3$-manifold $M$, with $\mathcal{F}$ transverse to $\partial M$. Let $S\subset M$ be a properly embedded oriented surface such that each component of $\partial S$ is either a leaf of or transverse to $\mathcal{F}\cap \partial M$. Then the f with $e(\mathcal{F})\in H^2(M,\partial M;\mathop{\mathrm{\mathbb Z}}\nolimits)$ and $[S]\in H_2(M,\

Figures (11)

  • Figure 1: A $2$-dimensional Reeb component is a subset of leaves of the foliation covering an annulus. The boundary of the annulus is composed of leaves with nonparallel coorientation. The interior leaves are lines.
  • Figure 2: Local view of a surface decomposition close to a transverse intersection with a suture. On both left and right, the sutures are coloured in grey.
  • Figure 3: The local views of a branched surface close to a triple point $c$ (left) and at the intersection with $\partial M$ (right). Big arrows indicate a coherent choice of coorientations for the sectors. The small grey arrows indicate the maw vector field. In this case, the normal orientation of $\partial M$ is pointing out of $M$.
  • Figure 4: Local view of the fibered neighbourhood $N(\mathcal{B})$ of a branched surface $\mathcal{B}$ close to a triple point of the branching locus. The grey intervals are some fibers. The small shaded areas are part of the vertical boundary $N_v(\mathcal{B})$.
  • Figure 5: Local view of the branched surface $\mathcal{B}_{i+1}$ close to a transverse intersection between $S_{i+1}$ and $\partial _vN(\mathcal{B}_i)$.
  • ...and 6 more figures

Theorems & Definitions (38)

  • Theorem 1.1: Thurston's Inequality
  • Theorem 1.2
  • Definition 2.1: Sutured manifold
  • Definition 2.2: Taut foliations
  • Definition 2.3: Surface decompositions
  • Definition 2.4: Hierarchies
  • Theorem 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8: Branched surface
  • ...and 28 more