[Paper Review] Euler class of taut foliations and Dehn filling

Ying Hu|arXiv (Cornell University)|Dec 3, 2019

Geometric and Algebraic Topology33 references4 citations

TL;DR

This paper establishes a necessary and sufficient condition for the Euler class of a taut foliation transverse to the core of a Dehn filling on a $Χ$-homology solid torus to vanish, using the relative Euler class and a modular congruence condition. It applies this to show that many Dehn fillings of hyperbolic fibered manifolds—such as the pretzel knot exterior $P(-2,3,2r+1)$ for $r\geq3$—admit left-orderable fundamental groups via taut foliations with zero Euler class.

ABSTRACT

In this article, we study the Euler class of taut foliations on the Dehn fillings of a $\mathbb{Q}$-homology solid torus. We give a necessary and sufficient condition for the Euler class of a foliation transverse to the core of the filling solid torus to vanish. We apply this condition to taut foliations on Dehn fillings of hyperbolic fibered manifolds and obtain many new left-orderable Dehn filling slopes on these manifolds. For instance, we show that when $X$ is the exterior of a pretzel knot $P(-2,3,2r+1)$, $r\geq 3$, $π_1(X(α_n))$ is left-orderable for a sequence of positive slopes $α_n$ with $α_0 =2g-2$ and $α_n o 2g-1$. Lastly, we prove that given any $\mathbb{Q}$-homology solid torus, the set of slopes for which the corresponding Dehn fillings admit a taut foliation transverse to the core with zero Euler class is nowhere dense in $\mathbb{R}\cup \{\frac{1}{0}\}$.

Motivation & Objective

• To determine when the Euler class of a taut foliation transverse to the core of a Dehn filling on a $Χ$-homology solid torus vanishes.
• To apply this criterion to construct new examples of left-orderable Dehn fillings on hyperbolic fibered 3-manifolds.
• To show that the set of slopes admitting taut foliations with zero Euler class is nowhere dense in the rational slope sphere.
• To extend results on left-orderability to families of positive slopes approaching $2g-1$ for certain pretzel knots.

Proposed method

• Uses the relative Euler class $e_{\sigma}(\mathcal{F})$ associated with an outward-pointing section $\sigma$ along $\partial X$.
• Applies Theorem 1.4: the Euler class of the extended foliation $\widehat{\mathcal{F}}$ on $X(p/q)$ vanishes if and only if $aq \equiv 1 \pmod{p}$, where $a = e_{\sigma}(\mathcal{F})([F])$.
• Analyzes the structure of the set of slopes $\mathcal{S}_{X,\mu}$ for which such foliations exist with zero Euler class.
• Proves that the set of such slopes is nowhere dense in $\mathbb{R} \cup \{1/0\}$ by showing it is a countable union of nowhere dense sets.
• Applies the criterion to fibered manifolds, particularly knot exteriors, using the Thurston norm and genus $g$ of the fiber surface.
• Establishes a simplified condition for integral slopes: $e_{\sigma}(\mathcal{F})([F]) = \pm1$ depending on the sign of the slope $m$.

Experimental results

Research questions

• RQ1 Under what conditions does the Euler class of a taut foliation transverse to the core of a Dehn filling vanish?
• RQ2 Which Dehn filling slopes on hyperbolic fibered 3-manifolds admit taut foliations with zero Euler class?
• RQ3 How dense is the set of slopes admitting taut foliations with zero Euler class in the space of all slopes?
• RQ4 Can the left-orderability of the fundamental group be established via taut foliations with zero Euler class for sequences of slopes approaching $2g-1$?
• RQ5 What is the role of the relative Euler class and modular congruence in determining the vanishing of the Euler class?

Key findings

• The Euler class of a taut foliation transverse to the core of a Dehn filling on a $Χ$-homology solid torus vanishes if and only if $aq \equiv 1 \pmod{p}$, where $a = e_{\sigma}(\mathcal{F})([F])$.
• For the pretzel knot $P(-2,3,2r+1)$ with $r \geq 3$, the fundamental group of the Dehn filling $X(\alpha_n)$ is left-orderable for a sequence of positive slopes $\alpha_n$ with $\alpha_0 = 2g - 2$ and $\alpha_n \to 2g - 1$.
• The set of slopes for which Dehn fillings admit taut foliations transverse to the core with zero Euler class is nowhere dense in $\mathbb{R} \cup \{1/0\}$.
• For integral slopes $m$, the Euler class vanishes if $e_{\sigma}(\mathcal{F})([F]) = 1$ when $m > 0$ and $-1$ when $m < 0$.
• The result confirms that left-orderability can be achieved via taut foliations with zero Euler class on infinitely many slopes near $2g - 1$ for fibered manifolds.
• The construction applies to all $Χ$-homology solid tori, and the set of such slopes is shown to be a countable union of nowhere dense sets, hence nowhere dense overall.