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Transactions of the American Mathematical Society

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

 
 

 

Nonexistence of twists and surgeries generating exotic 4-manifolds


Author: Kouichi Yasui
Journal: Trans. Amer. Math. Soc. 372 (2019), 5375-5392
MSC (2010): Primary 57R55; Secondary 57R65, 57R17
DOI: https://doi.org/10.1090/tran/7696
Published electronically: July 2, 2019
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Abstract: It is well known that for any exotic pair of simply connected closed oriented 4-manifolds, one is obtained from the other by twisting a compact contractible submanifold via an involution on the boundary. By contrast, here we show that for each positive integer $ n$, there exists a simply connected closed oriented 4-manifold $ X$ such that for any compact (not necessarily connected) codimension zero submanifold $ W$ with $ b_1(\partial W)<n$, the set of all smooth structures on $ X$ cannot be generated from $ X$ by twisting $ W$ and varying the gluing map. As a corollary, we show that there exists no ``universal'' compact 4-manifold $ W$ such that for any simply connected closed 4-manifold $ X$, the set of all smooth structures on $ X$ is generated from a 4-manifold by twisting a fixed embedded copy of $ W$ and varying the gluing map. Moreover, we give similar results for surgeries.


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Additional Information

Kouichi Yasui
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan
Email: kyasui@ist.osaka-u.ac.jp

DOI: https://doi.org/10.1090/tran/7696
Keywords: 4-manifolds, smooth structures, corks, minimal genus functions, Stein 4-manifolds
Received by editor(s): February 25, 2018
Received by editor(s) in revised form: August 31, 2018
Published electronically: July 2, 2019
Additional Notes: The author was partially supported by JSPS KAKENHI Grant Numbers 16K17593, 26287013 and 17K05220.
Article copyright: © Copyright 2019 American Mathematical Society