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Rigidity of the mod 2 families Seiberg–Witten invariants and topology of families of spin 4-manifolds

Part of: Differential topology

Published online by Cambridge University Press:  15 April 2021

Tsuyoshi Kato ,
Hokuto Konno  and
Nobuhiro Nakamura
Tsuyoshi Kato
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto606-8502, Japantkato@math.kyoto-u.ac.jp
Hokuto Konno
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo153-8914, Japankonno@ms.u-tokyo.ac.jp
Nobuhiro Nakamura
Affiliation:
Department of Mathematics, Osaka Medical College, 2-7 Daigaku-machi, Takatsuki City, Osaka569-8686, Japanmat002@osaka-med.ac.jp
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Abstract

We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of $4$-manifolds $M$ for which the inclusion maps $\operatorname {Diff}(M) \hookrightarrow \operatorname {Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin $4$-manifolds.

Keywords

Seiberg–Witten equationsdiffeomorphism grouphomeomorphism group

MSC classification

Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants
Secondary: 57R50: Diffeomorphisms
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 4 , April 2021 , pp. 770 - 808
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Copyright
© The Author(s) 2021

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