SMOOTHLY EMBEDDING SEIFERT FIBERED SPACES IN S4

Using an obstruction based on Donaldson's theorem, we derive strong restrictions on when a Seifert fibered space Y = F(e; p1/q1,....pk/qk) over an orientable base surface F can smoothly embed in S-4. This allows us to classify precisely when Y smoothly embeds provided e > k/2, where e is the normalized central weight and k is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant (u) over bar, we make some conjectures concerning Seifert fibered spaces which embed in S-4. Finally, we also provide some applications to doubly slice Montesinos links, including a classification of the smoothly doubly slice odd pretzel knots up to mutation.