Smoothly embedding Seifert fibered spaces in $S^4$

Abstract

Using an obstruction based on Donaldson's theorem, we derive strong restrictions on when a Seifert fibered space $Y=F(e;\frac{p_1}{q_1},\dots ,\frac{p_k}{q_k})$ 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 $\overline{\mu }$, 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.

35 pages, 14 figures. Comments welcome!