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

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          Abstract

          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.

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          Journal
          2014-01-29
          2014-12-05
          1401.7726

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          Custom metadata
          57M25, 57M50, 57M99
          53 pages. Updated introduction, added Problem 1.11 and Question 1.13. The statement and proof of Lemma 6.12 are also changed to address a gap in the proof of its original statement
          math.GT

          Geometry & Topology

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