Knotted surfaces in $4$-manifolds by knot surgery and stabilization
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- by Hee Jung Kim PDF
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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 $S^2\widetilde {\times }S^2$.References
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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
- 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8405-8427
- MSC (2010): Primary 57R40, 57R50; Secondary 57Q60
- DOI: https://doi.org/10.1090/tran/7450
- MathSciNet review: 3955551