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

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.