• Corpus ID: 119272202

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}
}
  • S. Boyer, A. Clay
  • Published 8 October 2015
  • Mathematics
  • arXiv: Geometric Topology
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… 

L-spaces, taut foliations, and graph manifolds

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

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

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

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

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

References

SHOWING 1-10 OF 13 REFERENCES

Graph manifold ℤ‐homology 3‐spheres and taut foliations

We show that a graph manifold which is a Z ‐homology 3 ‐sphere not homeomorphic to either S3 or Σ(2,3,5) admits a horizontal foliation. This combines with known results to show that the conditions of

Graph manifolds, left-orderability and amalgamation

We show that every irreducible toroidal integer homology sphere graph manifold has a left-orderable fundamental group. This is established by way of a specialization of a result due to Bludov and

On L-spaces and left-orderable fundamental groups

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

Ordered groups, eigenvalues, knots, surgery and L-spaces

  • A. ClayD. Rolfsen
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2011
Abstract We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined,

Holomorphic disks and genus bounds

We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the

Foliations transverse to fibers of Seifert manifolds

In this paper we prove the conjecture of Jankins and Neumann [JN2] about rotation numbers of products of circle homeomorphisms, which together with other results of [EHN] and [JN2] (mentioned below)

Foliations and the Geometry of 3-Manifolds

Preface 1. Surface bundles 2. The topology of S1 3. Minimal surfaces 4. Taut foliations 5. Finite depth foliations 6. Essential laminations 7. Universal circles 8. Constructing transverse laminations

Left-orderable fundamental groups and Dehn surgery

There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive;

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not