Veering triangulations and transverse foliations

Abstract

We present a combinatorial approach to the existence of foliations and contact structures transverse to a given pseudo-Anosov flow. Let $\varphi$ be a transitive pseudo-Anosov flow on a closed oriented 3-manifold. Our main technical result is that every codimension 1 foliation transverse to $\varphi$ is carried by a single branched surface coming from a veering triangulation. Combined with recent breakthrough work of Massoni, this reduces the existence problem for transverse foliations to something like the feasibility of a system of inequalities (rather than equations!) over $Homeo_+([0,1])$. As a proof of concept, we show that for the hyperbolic, fibered, non-L-space knot $10_{145}$, the natural pseudo-Anosov flow on the slope $s$ Dehn surgery admits a transverse foliation for $s\in (-\infty, 3)$, but does not admit such a foliation for $s\in [5,\infty)$. The negative result is part of a more general Milnor--Wood type phenomenon which puts limitations on some well known methods for constructing taut foliations on Dehn surgeries.

Figures (9)

• Figure 1: $\partial D^2 \times S^1$ shown on the left. On the right, we unfolded $\partial D^2 \times S^1$ to show the induced foliation. The regions corresponding with $\rho(\gamma_1)$ and $\rho(\gamma_1)^{-1}$ are labelled with $A$ and $A^{-1}$.
• Figure 2: An edge $e$ of $\tau^{(2)}$ with $n_e=2$ and $m_e=3$.
• Figure 3: The foliation on the dual 2-cell $e^*$.
• Figure 4: In the first two cases, $r_1 \leq_{\mathcal{R}} r_2$. In the last case, $r_1$ and $r_2$ are not comparable.
• Figure 5: (a) shows the restriction of $\widehat{\mathcal{F}}$ to a flow cylinder, and (b) shows more realistic geometry of the same flow cylinder.

Theorems & Definitions (32)

• Theorem A
• Proposition 1.3
• Theorem B
• Theorem C
• Theorem 1.4: Thurston, Brittenham brittenham.EssentialLaminationsSeifertfibered
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3