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Transactions of the American Mathematical Society

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

 
 

 

Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot


Authors: Matthew Hedden, Min Hoon Kim, Thomas E. Mark and Kyungbae Park
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 57M25, 57M27, 57R58, 57R65
DOI: https://doi.org/10.1090/tran/7786
Published electronically: May 20, 2019
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Abstract: We give two infinite families of examples of closed, orientable, irreducible 3-manifolds $ M$ such that $ b_1(M)=1$ and $ \pi _1(M)$ has weight 1, but $ M$ is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl, and Wilton and provides the first examples of irreducible manifolds with $ b_1=1$ that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.


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Additional Information

Matthew Hedden
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: mhedden@math.msu.edu

Min Hoon Kim
Affiliation: School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
Email: kminhoon@kias.re.kr

Thomas E. Mark
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Email: tmark@virginia.edu

Kyungbae Park
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
Email: kyungbaepark@snu.ac.kr

DOI: https://doi.org/10.1090/tran/7786
Received by editor(s): July 9, 2018
Received by editor(s) in revised form: December 18, 2018
Published electronically: May 20, 2019
Additional Notes: The first author’s work on this project was partially supported by NSF CAREER grant DMS-1150872, DMS-1709016, and an NSF postdoctoral fellowship.
The second author was partially supported by the POSCO TJ Park Science Fellowship.
The third author was supported in part by a grant from the Simons Foundation (523795, TM).
The fourth author was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF, F2018R1C1B6008364).
Kyungbae Park is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society