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

@article{Boyer2015SlopeDF, title={Slope detection, foliations in graph manifolds, and L-spaces}, author={Steven D. Boyer and Adam Clay}, journal={arXiv: Geometric Topology}, year={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 foliation. In this paper we extend the gluing theorem of arXiv:1401.7726 to graph manifold rational homology solid tori and use this to show that there are smooth foliations on the pieces of $W$ which come close to matching up on its JSJ tori. This is applied to prove that a graph manifold with left…

## 6 Citations

### L-spaces, taut foliations, and graph manifolds

- MathematicsCompositio Mathematica
- 2020

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…

### Links of rational singularities, L-spaces and LO fundamental groups

- Mathematics
- 2015

We prove that the link of a complex normal surface singularity is an L-space if and only if the singularity is rational. This via a result of Hanselman et al. (Taut foliations on graph manifolds,…

### On $1$-bridge braids, satellite knots, the manifold $v2503$ and non-left-orderable surgeries and fillings

- Mathematics
- 2020

We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot $K$ with property (D) with slope $\frac{p}{q}\ge 2g(K)-1$ is not…

### An Elementary Approach on Left-Orderability, Cables of Torus Knots and Dehn Surgery

- Mathematics
- 2016

Motivated by Clay and Watson's question on left-orderability of the fundamental group of the resultant space of an $r'$-surgery on the $(p, q)$-cable knots for $r' \in (pq-p-q,pq)$, this paper proves…

### Taut foliations, positive 3‐braids, and the L‐space conjecture

- MathematicsJournal of Topology
- 2020

We construct taut foliations in every closed 3‐manifold obtained by r ‐framed Dehn surgery along a positive 3‐braid knot K in S3 , where r<2g(K)−1 and g(K) denotes the Seifert genus of K . This…

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