Foliations, orders, representations, L-spaces and graph manifolds

By S. Boyer, Adam Clay. 2014

70 citations recorded by the source corpus.

We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology $3$-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions - using representations, using left-orders, using foliations, and using Heegaard-Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next we provide necessary and sufficient conditions for the existence of a co-oriented taut foliation on a graph manifold rational homology $3$-sphere, respectively a left-order on its fundamental group, which depend solely on families of detected slopes on the boundaries of its pieces. The fact that Heegaard-Floer methods can be used to detect families of slopes on the boundary of a Seifert fibred manifold combines with certain conjectures in the literature to suggest an L-space gluing theorem for rational homology $3$-sphere graph manifolds as well as other interesting problems in Heegaard-Floer theory.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/ebae6a274f268765937ee93cb5c47e07206f55b2

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Citations in the corpus

Floer simple manifolds and L-space intervals (2015, unknown): “To further explore the relationship of the above properties, Boyer and Clay [6] studied how each of these properties glue together when one splices together Seifert fibered spaces along the toroidal boundaries of fiber complements to form a graph manifold.”
Approximating C0-foliations by contact structures (2015, methodology): “smooth foliation by planes, say F′, by Imanishi [15]. Hence TF ≃ TF′ and the result follows from the C2-case. A further application of Theorem 1.2 is given by the following result, which was noted in [3], where the argument was not complete due to the absence of Theorem 1.2. Corollary 9.2. Let Mbe a”
Bordered Floer homology for manifolds with torus boundary via immersed curves (2016, methodology): “ knots in the three-sphere; see Gordon and Lidman [10, 11] and Boileau, Boyer, and Gordon [5]. A weaker version of the L-space gluing theorem was the key ingredient, along with work of Boyer and Clay [6], in proving the L-space conjecture for graph manifolds [15] (an alternate, constructive, proof i”