Smoothly embedding Seifert fibered spaces in 𝑆⁴

Issa, Ahmad; McCoy, Duncan

doi:10.1090/tran/8095

Smoothly embedding Seifert fibered spaces in $S^4$
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by Ahmad Issa and Duncan McCoy PDF

Trans. Amer. Math. Soc. 373 (2020), 4933-4974

Abstract:

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

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