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On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class

Steven Boyer, Cameron McA. Gordon, Ying Hu, Duncan McCoy

TL;DR

The paper investigates whether a closed orientable 3-manifold with a co-orientable taut foliation must admit one with vanishing Euler class, constructing infinite families of rational homology spheres with taut foliations but nonzero Euler class and examining the implications for left-orderability and the L-space conjecture. It combines Dehn surgery techniques on knots in $S^3$ with Heegaard Floer obstructions to show that large-even-denominator surgeries can yield taut foliations without zero Euler class, and develops a Seifert-fibration framework to compute and characterize the Euler class of the normal bundle $\nu_M$, including explicit necessary and sufficient conditions for $e(\nu_M) = 0$. The authors provide two perspectives—the knot-theoretic surgery approach and the discrete Fuchsian representation approach—to determine when $e(\nu_M)$ vanishes, showing the latter recovers the same vanishing criteria via lifting obstructions to $\widetilde{PSL}_2(\mathbb{R})$ and relates this to horizontal foliations. The work yields many Seifert fibred and toroidal examples with taut foliations but nonvanishing Euler class, clarifies the relationship between universal circle actions and left-orderable groups, and advances understanding of the L-space conjecture by presenting explicit counterexamples where left-orderability does not arise from universal circle lifts.

Abstract

In this article, we construct infinitely many (small Seifert fibred, hyperbolic and toroidal) rational homology $3$-spheres that admit co-orientable taut foliations, but none with vanishing Euler class. In the context of the $L$-space conjecture, these examples provide rational homology $3$-spheres that admit co-orientable taut foliations (and hence are not $L$-spaces) and have left-orderable fundamental groups, yet none of the left orders arise directly from the universal circle actions associated to co-orientable taut foliations. The hyperbolic and non-Seifert toroidal examples are obtained from Dehn surgeries on knots in the $3$-sphere and use Heegaard Floer homology to obstruct the existence of a co-orientable foliation with vanishing Euler class. For the Seifert fibred case, we establish necessary and sufficient conditions for the Euler class of the normal bundle of the Seifert fibration to vanish. Moreover, when the base orbifold is hyperbolic, we also provide a second proof of this condition from the viewpoint of discrete faithful representations of Fuchsian groups.

On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class

TL;DR

The paper investigates whether a closed orientable 3-manifold with a co-orientable taut foliation must admit one with vanishing Euler class, constructing infinite families of rational homology spheres with taut foliations but nonzero Euler class and examining the implications for left-orderability and the L-space conjecture. It combines Dehn surgery techniques on knots in with Heegaard Floer obstructions to show that large-even-denominator surgeries can yield taut foliations without zero Euler class, and develops a Seifert-fibration framework to compute and characterize the Euler class of the normal bundle , including explicit necessary and sufficient conditions for . The authors provide two perspectives—the knot-theoretic surgery approach and the discrete Fuchsian representation approach—to determine when vanishes, showing the latter recovers the same vanishing criteria via lifting obstructions to and relates this to horizontal foliations. The work yields many Seifert fibred and toroidal examples with taut foliations but nonvanishing Euler class, clarifies the relationship between universal circle actions and left-orderable groups, and advances understanding of the L-space conjecture by presenting explicit counterexamples where left-orderability does not arise from universal circle lifts.

Abstract

In this article, we construct infinitely many (small Seifert fibred, hyperbolic and toroidal) rational homology -spheres that admit co-orientable taut foliations, but none with vanishing Euler class. In the context of the -space conjecture, these examples provide rational homology -spheres that admit co-orientable taut foliations (and hence are not -spaces) and have left-orderable fundamental groups, yet none of the left orders arise directly from the universal circle actions associated to co-orientable taut foliations. The hyperbolic and non-Seifert toroidal examples are obtained from Dehn surgeries on knots in the -sphere and use Heegaard Floer homology to obstruct the existence of a co-orientable foliation with vanishing Euler class. For the Seifert fibred case, we establish necessary and sufficient conditions for the Euler class of the normal bundle of the Seifert fibration to vanish. Moreover, when the base orbifold is hyperbolic, we also provide a second proof of this condition from the viewpoint of discrete faithful representations of Fuchsian groups.

Paper Structure

This paper contains 15 sections, 16 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rational homology spheres admitting no taut foliations with vanishing Euler class
  3. Euler class of the normal bundle of a Seifert fibration
  4. Examples from surgeries on knots in the $3$-sphere
  5. The Euler class of the normal bundle of a Seifert fibration
  6. Seifert notation
  7. Proof of Corollary \ref{['cor: necessary condition']}
  8. Proof of Theorem \ref{['thm: Euler class zero']}
  9. Discrete representations of Fuchsian groups and the normal bundle of a Seifert fibration
  10. The second cohomology group of $\Gamma$
  11. The normal bundle of the Seifert fibring of $M$ and discrete representations
  12. Seifert fibred examples
  13. $\mathcal{B}$ is spherical
  14. $\mathcal{B}$ is Euclidean
  15. $\mathcal{B}$ is hyperbolic

Key Result

Theorem 1.2

Let $M$ be obtained by $p/q$-surgery on a knot $K$ in the $3$-sphere. If $q$ is even and $|p/q| > 2g(K) - 1$, then $M$ does not admit a co-orientable taut foliation with Euler class zero.

Figures (1)

  • Figure 1: The generators $\bar{x}_i$ in the presentation of $\Gamma = \pi_1(\mathcal{B})$

Theorems & Definitions (30)

  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10
  • Theorem 1.11
  • ...and 20 more