  In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a
foliation has Euler characteristic equal, up to sign, to the Euler class of the
foliation evaluated on [S], the homology class represented by S. The main
result of this paper is a converse for taut foliations: if the Euler class of a
taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler
characteristic of S and the underlying manifold is hyperbolic, then there
exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a
union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to
that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the
same Euler class.
  In the same paper Thurston proved that taut foliations on closed hyperbolic
3-manifolds have Euler class of norm at most one, and conjectured that,
conversely, any integral cohomology class with norm equal to one is the Euler
class of a taut foliation. This is the second of two papers that together give
a negative answer to Thurston's conjecture. In the first paper, counterexamples
were constructed assuming the main result of this paper.
