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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Smoothly embedding Seifert fibered spaces in $ S^4$


Authors: Ahmad Issa and Duncan McCoy
Journal: Trans. Amer. Math. Soc. 373 (2020), 4933-4974
MSC (2010): Primary 57M27, 57R40
DOI: https://doi.org/10.1090/tran/8095
Published electronically: March 31, 2020
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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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Additional Information

Ahmad Issa
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
Email: aissa@math.ubc.ca

Duncan McCoy
Affiliation: Départment de Mathématiques, Université du Québec à Montréal, Montréal, Québec, Canada
Email: duncan.mccoy@cirget.ca

DOI: https://doi.org/10.1090/tran/8095
Received by editor(s): April 11, 2019
Received by editor(s) in revised form: November 5, 2019
Published electronically: March 31, 2020
Article copyright: © Copyright 2020 by the authors