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Transactions of the American Mathematical Society

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

 
 

 

Knotted surfaces in $ 4$-manifolds by knot surgery and stabilization


Author: Hee Jung Kim
Journal: Trans. Amer. Math. Soc. 371 (2019), 8405-8427
MSC (2010): Primary 57R40, 57R50; Secondary 57Q60
DOI: https://doi.org/10.1090/tran/7450
Published electronically: February 27, 2019
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Abstract: Given a simply-connected closed $ 4$-manifold $ X$ and a smoothly embedded oriented surface $ \Sigma $, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in $ X$ that are pairwise homeomorphic to $ \Sigma $ but not diffeomorphic. We prove that for all known examples of surface knots constructed from knot surgery operations that preserve the fundamental group of the complement of surface knots, they become pairwise diffeomorphic after stabilizing by connected summing with one $ S^2\widetilde {\times }S^2$. When $ X$ is spin, we show in addition that any surfaces obtained by a knot surgery whose complements have cyclic fundamental group become pairwise diffeomorphic after one stabilization by $ S^2\widetilde {\times }S^2$.


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Additional Information

Hee Jung Kim
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 790-784, Republic of Korea
Email: heejungorama@gmail.com

DOI: https://doi.org/10.1090/tran/7450
Received by editor(s): January 12, 2017
Received by editor(s) in revised form: September 28, 2017
Published electronically: February 27, 2019
Additional Notes: The author was supported by NRF grant 2015R1D1A1A01059318 and BK21 PLUS SNU Mathematical Sciences Division.
Article copyright: © Copyright 2019 American Mathematical Society