L–spaces, taut foliations and the Whiteheadlink

We prove that if M is a rational homology sphere that is a Dehn surgery on the Whitehead link, then M is not an L–space if and only if M supports a coorientable taut foliation. The left orderability of some of these manifolds is also proved, by determining which of the constructed taut foliations have vanishing Euler class. We also present some more general results about the structure of the L–space surgery slopes for links with two unknotted components and linking number zero, and about the existence of taut foliations on the fillings of a k –holed torus bundle over the circle with some prescribed monodromy. Our results, combined with some results of Roberts, Shareshian and Stein (2003), also imply that all the rational homology spheres that arise as integer surgeries on the Whitehead link satisfy the L–space conjecture.