L-spaces, taut foliations and fibered hyperbolic two-bridge links

Abstract: We prove that if $M$ is a rational homology sphere that is Dehn surgery on a fibered hyperbolic two-bridge link, then $M$ is not an $L$-space if and only if $M$ supports a coorientable taut foliation. As a corollary we show that if $K'$ is obtained by a non-trivial knot $K$ as result of an operation called two-bridge replacement, then all non-meridional surgeries on $K'$ support coorientable taut foliations. This operation generalises Whitehead doubling and as a particular case we deduce that all non-meridional surgeries on Whitehead doubles of a non-trivial knot support coorientable taut foliations.