  Under suitable hypotheses, we prove that a form of a projective homogeneous
variety $G/P$ defined over the function field of a surface over an
algebraically closed field has a rational point. The method uses an
algebro-geometric analogue of simple connectedness replacing the unit interval
by the projective line. As a consequence, we complete the proof of Serre's
Conjecture II in Galois cohomology for function fields over an algebraically
closed field.
