# L-spaces, taut foliations, and graph manifolds

@article{Hanselman2020LspacesTF, title={L-spaces, taut foliations, and graph manifolds}, author={Jonathan Hanselman and Jacob Rasmussen and Sarah Dean Rasmussen and Liam Watson}, journal={Compositio Mathematica}, year={2020}, volume={156}, pages={604 - 612} }

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

## 29 Citations

L-spaces, taut foliations and the Whitehead link

- Mathematics
- 2022

We prove that if M is a rational homology sphere that is a Dehn surgery on the Whitehead link, then M is not an L-space if and only if M supports a coorientable taut foliation. The left orderability…

Floer homology, group orderability, and taut
foliations of hyperbolic 3–manifolds

- Mathematics
- 2019

This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and…

Euler class of taut foliations and Dehn filling

- Mathematics
- 2019

We study the Euler class of co-orientable taut foliations on rational homology spheres. Given a rational homology solid torus $X$, we give necessary and sufficient conditions for the Euler class of…

The equivalence of lattice and Heegaard Floer homology

- Mathematics
- 2021

We prove that if Y is a 3-manifold which is the boundary of a plumbing of a tree of disk bundles over S, then the lattice homology of Y coincides with the Heegaard Floer homology of Y . We also give…

Floer homology and covering spaces

- MathematicsGeometry & Topology
- 2018

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a…

Thin links and Conway spheres

- Mathematics
- 2020

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal δ-grading. This leads to the broader class of thin links that one would like to…

Orderability and Dehn filling

- Mathematics
- 2016

Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental…

On the set of L-space surgeries for links

- MathematicsAdvances in Mathematics
- 2018

Slope detection and toroidal 3-manifolds

- Mathematics
- 2021

We investigate the L-space conjecture for toroidal 3-manifolds using various notions of slope detection. This leads to a proof that toroidal 3-manifolds with small order first homology have…

## References

SHOWING 1-10 OF 42 REFERENCES

Slope detection, foliations in graph manifolds, and L-spaces

- Mathematics
- 2015

A graph manifold rational homology $3$-sphere $W$ with a left-orderable fundamental group admits a co-oriented taut foliation, though it is unknown whether it admits a smooth co-oriented taut…

Approximating C0-foliations by contact structures

- Mathematics
- 2015

We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C0-approximated by both positive and negative contact structures…

L-space intervals for graph manifolds and cables

- MathematicsCompositio Mathematica
- 2017

We present a graph manifold analog of the Jankins–Neumann classification of Seifert fibered spaces over $S^{2}$ admitting taut foliations, providing a finite recursive formula to compute the L-space…

On L-spaces and left-orderable fundamental groups

- Mathematics
- 2011

Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions…

$C^0$ Approximations of foliations

- Mathematics
- 2015

Suppose that $\mathcal F$ is a transversely oriented, codimension one foliation of a connected, closed, oriented 3-manifold. Suppose also that $\mathcal F$ has continuous tangent plane field and is…

Approximating $C^{1,0}$-foliations

- Mathematics
- 2014

We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented…

Foliations and the topology of 3-manifolds

- Mathematics
- 1987

In this announcement we discuss the close relationship between the topology of 3-manifolds and the foliations that is possesses. We will introduce and state the main result, then use it and the ideas…

Bordered Heegaard Floer Homology and Graph Manifolds

- Mathematics
- 2013

We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type D trimodule associated to the trivial S^1 bundle over the pair of pants P. The second is a bimodule…